PRIMARY OBJECTIVE:
      I. To determine if the invasive disease-free survival (iDFS) with T-DM1 and tucatinib is
      superior to the iDFS in the control arm (T-DM1 + placebo) when administered to high risk
      patients with HER2-positive breast cancer and residual disease after neoadjuvant
      HER2-directed therapy.
      SECONDARY OBJECTIVES:
      I. To evaluate whether treatment with tucatinib plus T-DM1 compared to treatment with T-DM1
      alone (T-DM1 plus placebo) improves the following:
      Ia. Breast cancer free survival (BCFS). Ib. Distant recurrence-free survival (DRFS). Ic.
      Brain metastases-free survival (BMFS). Id. Overall survival (OS). II. To evaluate whether
      treatment with tucatinib plus T-DM1 compared to treatment with T-DM1 alone (T-DM1 plus
      placebo) reduces the incidence of brain metastases.
      OUTLINE: Patients are randomized to 1 of 2 arms.
      ARM I: Patients receive T-DM1 intravenously (IV) over 30-90 minutes on day 1 and placebo
      orally (PO) twice daily (BID) on days 1-21. Treatment repeats every 21 days for up to 14
      cycles in the absence of disease progression or unacceptable toxicity.
      ARM II: Patients receive T-DM1 IV over 30-90 minutes on day 1 and tucatinib PO BID on days
      1-21. Treatment repeats every 21 days for up to 14 cycles in the absence of disease
      progression or unacceptable toxicity.
      After completion of study treatment, patients are followed up at 30 days, then every 6 months
      for 10 years.
    
        Inclusion Criteria:
          -  HER2-positive status will be based on pretreatment biopsy material and defined as an
             immunohistochemistry (IHC) score of 3+ and/or positive by in situ hybridization (ISH)
             according to current American Society of Clinical Oncology (ASCO) College of American
             Pathologists (CAP) guidelines. Central testing is not required
             * Known hormone receptor (HR) status as defined by ASCO/CAP guidelines. Hormone
             receptor positive status can be determined by either known positive estrogen receptor
             (ER) or known positive progesterone receptor (PR) status; hormone receptor negative
             status must be determined by both known negative ER and known negative PR
          -  Patients with residual HR-negative, HER2 positive (+) disease in the breast and/or
             lymph nodes per the surgical pathology report are eligible; however, patients with HR+
             HER2+ cancers must have node-positive residual disease per the surgical pathology
             report in order to qualify for the study. The presence of residual invasive disease in
             the breast is not mandatory for these patients. Note: The presence of micrometastases
             in lymph nodes after preoperative therapy counts as residual disease, whereas the
             presence of isolated tumor cells does not
          -  Patients with synchronous bilateral invasive disease are eligible provided both
             lesions were confirmed to be HER2-positive, and at least one of the lesions meets the
             criteria outlined above. Multifocal disease is allowed, as long as the largest
             biopsied breast tumor was HER2-positive
          -  Patients with clinical stage T1-4, N0-3 disease at presentation and residual invasive
             disease postoperatively as defined above are eligible. (Note: Patients with T1a/bN0
             tumors are not eligible)
          -  Patients must have received neoadjuvant chemotherapy with one of the following
             regimens: docetaxel/trastuzumab/pertuzumab (THP), paclitaxel/methotrexate/cisplatin
             (TMP), doxorubicin/cyclophosphamide/paclitaxel/trastuzumab/pertuzumab (AC-TH(P));
             docetaxel/carboplatin/trastuzumab/pertuzumab (TCH(P));
             fluorouracil/doxorubicin/cyclophosphamide-docetaxel/trastuzumab/pertuzumab
             (FAC-TH(P)), or
             fluorouracil/epirubicin/cyclophosphamide-docetaxel/trastuzumab/pertuzumab (FEC-TH(P)).
             Note: apart from TCHP, where T is docetaxel, treatment with docetaxel or paclitaxel is
             acceptable
          -  Patients may have received =< 2 cycles of T-DM1 in the adjuvant setting. Note: These
             patients will be randomized to receive a further 14 cycles of T-DM1 and
             tucatinib/placebo as tolerated. The most recent cycle of T-DM1 should have been
             administered =< 5 weeks prior to registration
             * N.B: Both of the following two criteria need to be met for the patient to be
             eligible for this study
               -  An interval of no more than 12 weeks between the completion date of the last
                  definitive treatment (e.g. postoperative chemotherapy or radiation, or if neither
                  given, breast surgical date) and the date of registration
               -  Patients must be registered on study within =< 180 days of the date of the most
                  recent definitive breast cancer surgery (not including reconstructive surgery)
          -  All systemic chemotherapy should have been completed preoperatively unless
             participating in EA1181 (CompassHER2 pathologic complete response [pCR])
          -  Toxicities related to prior systemic treatment should have resolved or be at baseline,
             apart from alopecia and peripheral neuropathy =< grade 1
          -  Adequate excision: surgical removal of all clinically evident disease in the breast
             and lymph nodes as follows:
               -  Breast surgery: total mastectomy with no gross residual disease at the margin of
                  resection, or breast-conserving surgery with histologically negative margins of
                  excision
               -  For patients who undergo breast-conserving surgery, the margins of the resected
                  specimen must be histologically free of invasive tumor and ductal carcinoma in
                  situ (DCIS) as determined by the local pathologist. If pathologic examination
                  demonstrates tumor at the line of resection, additional operative procedures may
                  be performed to obtain clear margins. If tumor is still present at the resected
                  margin after re-excision(s), the patient must undergo total mastectomy to be
                  eligible. Patients with margins positive for classic lobular carcinoma in situ
                  (LCIS) are eligible without additional resection
               -  Lymph node surgery ** The axilla needs to be evaluated with either sentinel node
                  biopsy or axillary lymph node dissection. If patients have a sentinel lymph node
                  biopsy and sentinel nodes are negative, no further axillary treatment is
                  necessary. If patients have isolated tumor cells (ITCs) in the setting of
                  residual breast disease, at least one of the following is required: axillary
                  lymph node dissection (ALND) or planned nodal irradiation. If patients have
                  micro- or macro-metastatic nodal disease, an ALND is required
          -  Eastern Cooperative Oncology Group (ECOG) performance status 0-1
          -  Absolute neutrophil count (ANC) >= 1,000/mm^3
          -  Hemoglobin >= 8 g/dL (Note: packed red blood cells [PRBC] transfusion is not permitted
             to achieve eligibility)
          -  Platelet count >= 100,000/mm^3
          -  Creatinine =< 1.5 x upper limit of normal (ULN)
          -  Total bilirubin =< 1.5 x upper limit of normal (ULN) or direct bilirubin within the
             institutional normal range for patients with Gilbert's syndrome
          -  Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =< 2.5 x upper limit
             of normal (ULN)
          -  Screening left ventricular ejection fraction (LVEF) >= 50% on echocardiogram (ECHO) or
             multiple-gated acquisition (MUGA) after receiving neoadjuvant chemotherapy and no
             decrease in LVEF by more than 15% absolute points from the pre-chemotherapy LVEF. Or,
             if pre-chemotherapy LVEF was not assessed, the screening LVEF must be >= 55% after
             completion of neoadjuvant chemotherapy. Note: LVEF assessment may be repeated once up
             to 3 weeks following the initial screening assessment to assess eligibility
        Exclusion Criteria:
          -  Prior receipt of T-DM1 in the neoadjuvant setting is not allowed
               -  Systemic therapy must have consisted of at least 9 weeks of preoperative taxane
                  and trastuzumab (or Food and Drug Administration [FDA]-approved biosimilar) with
                  or without pertuzumab. Patients who have received at least 9 weeks of
                  preoperative taxane, pertuzumab and margetuximab are also eligible. Note:
                  Patients who complete at least nine of a planned twelve doses of weekly
                  paclitaxel, or three of a planned four doses of docetaxel, but discontinue
                  prematurely due to toxicity (i.e. peripheral neuropathy =< grade 1) are eligible.
                  Patients receiving dose-dense chemotherapy regimens are also eligible. Prior use
                  of nab-paclitaxel (Abraxane) instead of paclitaxel or docetaxel is permitted.
                  Prior use of subcutaneous trastuzumab (Hylecta) is also allowed
               -  Patients who received neoadjuvant systemic therapy which included experimental
                  HER2-targeted therapy/therapies are potentially eligible, as long as the
                  investigational agent was not a HER2-targeted antibody-drug conjugate (e.g.
                  T-DM1, DS-8201a [trastuzumab deruxtecan]) or a HER2 targeted tyrosine kinase
                  inhibitor (TKI) (e.g. tucatinib, lapatinib, neratinib)
          -  No adjuvant treatment with any anti-cancer investigational drug within 28 days prior
             to registration
          -  Not pregnant and not nursing, because this study involves an agent that has known
             genotoxic, mutagenic and teratogenic effects. Therefore, for women of childbearing
             potential only, a negative serum pregnancy test done =< 7 days prior to registration
             is required
          -  Patients with known active and/or untreated hepatitis B or hepatitis C or chronic
             liver disease are ineligible. Patients with a diagnosis of hepatitis B or C that has
             been treated and cleared and normal liver function are eligible to participate in the
             study if the other eligibility parameters are met
          -  Stage IV (metastatic) breast cancer
          -  History of any prior (ipsi- or contralateral) invasive breast cancer within 3 years of
             registration
          -  Patients with ER+ HER2+ residual invasive disease that is lymph node-negative per the
             surgical pathology report
          -  Evidence of recurrent disease following preoperative therapy and surgery
          -  Patients for whom radiotherapy would be recommended for breast cancer treatment but
             for whom it is contraindicated because of medical reasons (e.g., connective tissue
             disorder or prior ipsilateral breast radiation)
          -  History of exposure to the following cumulative doses of anthracyclines: doxorubicin >
             240 mg/m^2; epirubicin or liposomal doxorubicin-hydrochloride (Myocet) > 480 mg/m^2.
             For other anthracyclines, exposure equivalent to doxorubicin > 240 mg/m^2
          -  Cardiopulmonary dysfunction as defined by any of the following:
               -  History of National Cancer Institute (NCI) CTCAE version (v) 5.0 grade >= 3
                  symptomatic congestive heart failure (CHF) or New York Heart Association (NYHA)
                  criteria class >= II
               -  Angina pectoris requiring anti-anginal medication, serious cardiac arrhythmia not
                  controlled by adequate medication, severe conduction abnormality, or clinically
                  significant valvular disease
               -  High-risk uncontrolled arrhythmias: i.e., atrial tachycardia with a heart rate >
                  100/min at rest, significant ventricular arrhythmia (ventricular tachycardia) or
                  higher-grade atrioventricular block (AV)-block (second degree AV-block type 2
                  [Mobitz 2] or third degree AV-block)
               -  Significant symptoms (grade >= 2) relating to left ventricular dysfunction,
                  cardiac arrhythmia, or cardiac ischemia while or since receiving preoperative
                  therapy
               -  History of a decrease in left ventricular ejection fraction (LVEF) to < 40% with
                  prior trastuzumab treatment (e.g., during preoperative therapy)
               -  Uncontrolled hypertension (systolic blood pressure > 180 mmHg and/or diastolic
                  blood pressure > 100 mmHg)
          -  Current severe, uncontrolled systemic disease
          -  Major surgical procedure unrelated to breast cancer or significant traumatic injury
             within 28 days prior to registration or anticipation of the need for major surgery
             during the course of study treatment
          -  History of intolerance, including grade 3 to 4 infusion reaction or hypersensitivity
             to trastuzumab or murine proteins or any components of the product
          -  Peripheral neuropathy of any etiology that exceeds grade 1
          -  Assessment by the investigator as being unable or unwilling to comply with the
             requirements of the protocol
          -  Use of a strong CYP3A4 or CYP2C8 inhibitor within 2 weeks, or use of a strong CYP3A4
             or CYP2C8 inducer within 5 days prior to registration is prohibited. Please note that
             use of sensitive CYP3A substrates should be avoided two weeks before registration and
             during study treatment. Additionally, CYP3A4 or CYP2C8 inducers are prohibited as
             concomitant medications within 5 days following discontinuation of tucatinib treatment