Inclusion Criteria:

1. Age ≥18 years old.
2. Willing and able to give informed consent.
3. Diagnosed with chronic myelogenous leukemia (CML) in chronic phase and have either the b3a2 (e14a2) or b2a2 (e13a2) variants that give rise to the p210 BCR::ABL1 protein. Subtype classification whether b3a2 (e14a2) or b2a2 (e13a2) is not required for study eligibility.
4. Must have a documented history of attempting only one prior TKI discontinuation under the guidance of a treating physician. TKI includes dasatinib, imatinib or nilotinib.
5. Must have met all the following criteria prior to first attempt to discontinue their TKI:

* Stable molecular response (MR4; \< 0.01% IS) for \> 2 years (with allowance for a two-week variance), as documented on at least four tests, performed at least three months apart (e.g., If a patient has had \>4 PCR tests performed during the two years leading up to their initial TKI discontinuation, any value between 0.01 and 0.05% IS is considered a stable result, however, at least four tests must be \< 0.01% IS. If any results are \>0.05% IS, tests must have been repeated within one month and be less than 0.01% IS and stable.
* Treatment with one of the following FDA approved TKIs; imatinib, dasatinib, nilotinib at any dose for a minimum of approximately three years (allowance of a four-week variance) prior to discontinuing TKIs.
* Has been on any number of TKIs, but has not been resistant to any TKI (changes made for intolerance are allowed).
6. Must have relapsed (defined as loss of major molecular response (MMR), RQ-PCR for BCR::ABL1 \>0.1% IS after first attempted TKI discontinuation.
7. After first failed TFR attempt, must have a minimum duration of one year of retreatment with TKI, and must plan to remain on that TKI for a minimum of 12 months during the combination treatment phase.
8. Current TKI must be the same as the TKI being taken prior to the initial TFR attempt (e.g., if patient is on imatinib prior to first TFR attempt, they should be on imatinib at time of enrollment on this study).
9. Eastern Cooperative Oncology Group (ECOG) performance status 0-3.
10. Must have a RQ-PCR for BCR::ABL1 \< 0.0032% IS (MR4.5) reported by the trial designated central lab at the time of study enrollment.
11. Lipase ≤ 1.5 x upper limit of normal (ULN). For lipase \> ULN - ≤ 1.5 x ULN, value should be considered not clinically significant and not associated with risk factors for acute pancreatitis.
12. Female patients must meet one of the following:

* Postmenopausal for at least one year before the screening visit,
* Surgically sterile
* If they are of childbearing potential, agree to practice two effective methods of contraception from the time of signing of the informed consent form through 90 days after the last dose of study drug,
* Must also adhere to the guidelines of any treatment-specific pregnancy prevention program, if applicable
* Agree to practice true abstinence when this is in line with the preferred and usual lifestyle of the subject. (Periodic abstinence \[e.g., calendar, ovulation, symptothermal, postovulation methods\] and withdrawal are not acceptable contraception methods.)
13. Male patients, even if surgically sterilized (i.e., status post vasectomy), must agree to one of the following:

* Practice effective barrier contraception during the entire study treatment period and through 90 days after the last study drug dose.
* Must also adhere to the guidelines of any treatment-specific pregnancy prevention program, if applicable.
* Agree to practice true abstinence when this is in line with the preferred and usual lifestyle of the subject. (Periodic abstinence \[e.g., calendar, ovulation, symptothermal, postovulation methods\] and withdrawal are not acceptable methods of contraception.)

Exclusion Criteria:

1. History of accelerated or blast phase CML.
2. A second malignancy requiring active treatment.
3. History of recent (within 12 months) acute pancreatitis or chronic pancreatitis.
4. Subjects who have previously received treatment with asciminib.
5. Subjects with platelet (PLT) count \< 100 × 109/L or an absolute neutrophil count (ANC) of \< 1 × 109/L or hemoglobin \< 8 g/dL.
6. Aspartate aminotransferase (AST) and alanine transaminase (ALT) ≥3 times the institutional upper limit of normal.
7. Creatinine clearance \< 40 mL/min.
8. Total bilirubin ≥ 1.5 times the institutional upper limit of normal (unless direct bilirubin is within normal limits).
9. Pregnant or lactating.
10. Unable to comply with lab appointment schedule and patient-reported outcome (PRO) assessments.
11. Another investigational drug within four weeks of enrollment.
12. Any serious medical or psychiatric illness that could, in the investigator's opinion, interfere with the completion of treatment according to this protocol.
13. Patient has undergone a prior allogeneic stem cell transplant.
14. Screening 12-lead electrocardiogram (ECG) showing a baseline corrected QT interval \>480msec (patients with a pacemaker will still be eligible with QTc\>500msec).
15. Known active hepatitis B infection.

Eligibility for TFR Phase:

1. Stable molecular response (MR4.5; \< 0.0032% IS) documented on at least three tests (may include TFR phase screening PCR) by the trial designated lab, performed approximately three months apart while on combination phase.
2. TFR phase screening PCR RQ-PCR for BCR::ABL1 \< 0.0032% IS (MR4.5) by the trial designated lab.
3. ECOG 0-3.
4. Completion of 12 cycles on the combination therapy phase.