Inclusion Criteria:

• - Patients with a diagnosis of Philadelphia chromosome (Ph)- or BCR-ABL1-positive CML (as determined by cytogenetics, fluorescence in situ hybridization [FISH], or polymerase chain reaction [PCR]) - Patients should be receiving therapy with nilotinib or dasatinib, whether as initial therapy or after prior TKI, at a dose equal or lower than the standard dose.

• - Patients must have received TKI therapy for at least 24 months and not have required dose reductions, escalations, discontinuation or re-initiation after discontinuation of TKI in the last 6 months.

• - Patients must be in CCyR (by conventional karyotype or FISH, or BCR-ABL/ABL =< 1% IS if no cytogenetic analysis available within 3 months) - Patients must have detectable BCR-ABL1 transcript levels meeting at least one of the following criteria: a.

Patient has never achieved a MMR after 18 months of therapy with their current TKI, or b. Patient has not achieved MR4.5 after 36 months of therapy with their current TKI, or c. Patient has lost MMR or MR4.5 confirmed in at least two consecutive analyses separated by at least 1 month, or d. BCR-ABL1 transcript levels have reached a plateau defined as a ratio that is stable in a molecular response category (i.e., MMR, MR4 or MR4.5) in the last at least 12 months, with at least 3 values obtained during this period.

• - Patients must not have had a known continuous interruption of TKI therapy of greater than 14 days or for a total of 6 weeks in the 6 months prior to enrollment, unless the interruption was for an accident, unrelated hospitalization or surgical procedure, or for a treatment-free remission attempt that was unsuccessful and required re-initiation of therapy.

• - Eastern Cooperative Oncology Group (ECOG) performance status =< 2.

• - Creatinine =<1.5 x institutional upper limit of normal.

• - Amylase and lipase values =< 3.0 x institutional upper limit of normal.

• - Alkaline phosphatase =< 2.5 x institutional upper limit of normal unless considered to be not of hepatic origin.

• - Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT]) =< 3 x institutional upper limit of normal.

• - Total bilirubin =< 1.5 x institutional upper limit of normal (=< 3 x upper limit of normal in patients with known Gilbert's syndrome) - The effects of ABL001 on the developing human fetus are unknown.

For this reason, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation. Women of child-bearing potential must agree to use highly effective methods of contraception during dosing and for 30 days after study treatment. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Allowable methods of birth control: Total abstinence (when this is in line with the preferred and usual lifestyle of the subject). Periodic abstinence (e.g., calendar, ovulation, symptothermal, post-ovulation methods) and withdrawal are not acceptable methods of contraception. Female sterilization (have had surgical bilateral oophorectomy with or without hysterectomy), total hysterectomy, or tubal ligation at least six weeks before the start of study treatment. In case of oophorectomy alone, only when the reproductive status of the woman has been confirmed by follow up hormone level assessment. Male sterilization (at least 6 months prior to screening). The vasectomized male partner should be the sole partner for that subject. Use of oral, injected or implanted hormonal methods of contraception or placement of an intrauterine device (IUD) or intrauterine system (IUS), or other forms of hormonal contraception that have comparable efficacy (failure rate <1%), for example hormone vaginal ring or transdermal hormone contraception. Sexually active males must use a condom during intercourse while taking the drug and for 30 days after stopping treatment and should not father a child in this period. A condom is required to be used also by vasectomized men in order to prevent delivery of the drug via seminal fluid.

Exclusion Criteria:

• - Patients with New York Heart Association (NYHA) class III or IV congestive heart failure or left ventricular ejection fraction (LVEF) < 40% by echocardiogram or multigated acquisition (MUGA) scan.

• - Patients with a history of myocardial infarction within the last 6 months or unstable/uncontrolled angina pectoris or history of severe and/or uncontrolled ventricular arrhythmias.

Clinically significant cardiac arrhythmias (e.g., ventricular tachycardia), complete left bundle branch block, high-grade atrioventricular (AV) block (e.g., bifascicular block, Mobitz type II and third degree AV block)

• - Corrected QT interval (QTc) of > 480 milliseconds (ms) on baseline electrocardiogram (ECG) (using corrected QT interval per institutional standard) - Long QT syndrome, family history of idiopathic sudden death or congenital long QT syndrome, or any of the following: a.

Risk factors for Torsades de Pointes (TdP) including uncorrected hypokalemia or hypomagnesemia, history of cardiac failure, or history of clinically significant/symptomatic bradycardia. b. Concomitant medication(s) with a known risk to prolong the QT interval and/or known to cause Torsades de Pointes that cannot be discontinued or replaced 7 days prior to starting study drug by safe alternative medication.

• - Patients with known infection with human immunodeficiency virus (HIV) or active hepatitis B or C.

• - Patients with known conditions that would significantly affect the ingestion or gastrointestinal absorption of drugs administered orally.

• - Nursing women, women of childbearing potential (WOCBP) with positive blood or urine pregnancy test, or women of childbearing potential who are not willing to maintain adequate contraception.

• - History of acute pancreatitis within 1 year of study entry or past medical history of chronic pancreatitis.

• - Absolute neutrophil count (ANC) < 500/mm^3.

• - Platelet count < 50,000 mm^3.

• - History of other active malignancy within 2 years prior to study entry with the exception of previous or concomitant basal cell skin cancer and previous carcinoma in situ treated curatively.

• - Treatment with medications that meet one of the following criteria and that cannot be discontinued at least one week prior to the start of treatment with study treatment: Moderate or strong inducers of CYP3A.

Moderate or strong inhibitors of CYP3A and/or P-glycoprotein (P-gp). Substrates of CYP3A4/5, CYP2C8, or CYP2C9 with narrow therapeutic index.

• - Previous treatment with or known/ suspected hypersensitivity to ABL001 or any of its excipients.

- Subject has any other significant medical or psychiatric history that in the opinion of the investigator would adversely affect participation in this study

PRIMARY OBJECTIVE:

• I. To determine the clinical activity of the combination of asciminib (ABL001) and a tyrosine kinase inhibitor (TKI) in patients with chronic myeloid leukemia (CML) in complete cytogenetic response (CCyR) but detectable BCR-ABL1 transcript.

SECONDARY OBJECTIVES:

• I. To determine the effect of the combination of ABL001 and TKI on the rate of mismatch repair (MR)4, MR4.5, and sustained MR4.5.

• II. To investigate treatment-free remission after at least 2 years of sustained deep molecular remission.

• III. To determine the safety of the combination of asciminib and tyrosine kinase inhibitors.

• IV. To determine the event-free survival (EFS), survival free from transformation to accelerated and blast phase (TFS), and overall survival (OS).

EXPLORATORY OBJECTIVES:

• I. To determine the rate of minimal residual disease (MRD) clearance using droplet digital polymerase chain reaction (ddPCR) detecting the BCR-ABL1 fusion transcript.

• II. To determine the effect of therapy on quiescent leukemic Philadelphia chromosome positive (Ph+) stem cells (CFSEmax/CD34+).

• III. To determine the effect of this combination on mutations in ABL1 and mutations in clonal hematopoiesis of indeterminate potential (CHIP)-associated genes using molecular barcode sequencing.

• IV. To determine the effect of therapy on bone marrow progenitors in clonogenic assays.

• V. To describe immune effects of the combination of TKI and ABL001.

• VI. To describe patient reported outcomes (PRO) using the MD Anderson Symptom Inventory (MDASI)-CML instrument.

OUTLINE: Patients receive asciminib orally (PO) twice daily (BID) for up to 36 months while receiving standard of care dasatinib or nilotinib in the absence of disease progression or unacceptable toxicity. Patients may continue to receive asciminib after 36 months at the discretion of investigator. After completion of study treatment, patients are followed up every 4-8 weeks for 6 months and then every 3-6 months thereafter.

Arms

Experimental: Treatment (asciminib)

Patients receive asciminib PO BID for up to 36 months while receiving standard of care dasatinib or nilotinib in the absence of disease progression or unacceptable toxicity. Patients may continue to receive asciminib after 36 months at the discretion of investigator.

Interventions

Drug: - Asciminib

Given PO