Asciminib as Maintenance Treatment After Cellular Therapies for Adults With Philadelphia Chromosome Positive Acute Lymphoblastic Leukemia

Inclusion Criteria:

* PRE-SCREENING: Documented informed consent of the participant and/or legally authorized representative
* PRE-SCREENING: Age ≥ 18 years
* PRE-SCREENING: Participant was diagnosed with Ph+ ALL according to World Health Organization criteria. The BCR::ABL1 translocation may be detected by fluorescence in situ hybridization (FISH), polymerase chain reaction (PCR), next generation sequencing (NGS), or cytogenetics at least once any time prior to cellular therapy. Participants may have p190 or p220 isoform, and participants with T315I mutation are not excluded
* PRE-SCREENING: Participant meets one of the following criteria:

  * Arm 1: Have a date for HSCT scheduled within the next 30 days or have received HSCT within the last 30 days. Note: all HSCT donors, conditioning regimens, and GVHD prophylaxis regimens will be acceptable.
  * Arm 2: Have a date for CAR T cell infusion scheduled within the next 30 days or have received CAR T cell infusion within the last 30 days
* PRE-SCREENING: History of pancreatitis within 1 year of study entry or past medical history of chronic pancreatitis
* PRE-SCREENING: History of allergic reactions attributed to compounds of similar chemical or biologic composition to study agent
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Documented informed consent of the participant and/or legally authorized representative.

  * Assent, when appropriate, will be obtained per institutional guidelines
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Eastern Cooperative Oncology Group (ECOG) ≤ 2
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Participant is between day +30 and +150 after one of the following cellular therapies:

  * Arm 1: HSCT

    * Participants on Arm 1 must be fully engrafted post-HSCT.
  * Arm 2: CD19-targeted CAR T cell therapy (brexucabtagene autoleucel, tisagenlecleucel, obecabtagene autoleucel, investigational CD19 CAR T cell therapy)

    * Participants on Arm 2 must be fully recovered from cytopenia
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Day 30 (+/- 5 days) marrow post cellular therapy should show evidence of complete morphologic remission defined as \< 5% bone marrow (BM) blasts, no extramedullary disease, and transfusion independence
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Participant should have no morphological evidence of relapse
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Absolute neutrophil count (ANC) ≥ 500/mm\^3 for 3 days

  * NOTE: Patients are allowed growth factors
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Platelets ≥ 75,000/mm\^3
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Hemoglobin ≥ 9g/dL
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Total bilirubin ≤ 1.5 X upper limit of normal (ULN) (unless has Gilbert's disease)
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Aspartate aminotransferase (AST) ≤ 3.0 x ULN
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Alanine aminotransferase (ALT) ≤ 3.0 x ULN
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: serum creatinine \<1.5 mg/dL
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Left ventricular ejection fraction (LVEF) ≥ 50%

  * Note: To be performed within 28 days prior to day 1 of protocol therapy
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Bazett's corrected QT interval (QTcB) ≤ 480 ms

  * Note: To be performed within 28 days prior to day 1 of protocol therapy
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Oxygen (O2) saturation \> 90% on room air
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Seronegative for HIV antigen (Ag)/antibody (Ab) combination (combo), hepatitis C virus (HCV), active hepatitis B virus (HBV) (surface antigen negative) OR

  * If seropositive for HIV, HCV or HBV, nucleic acid quantitation must be performed. Viral load must be undetectable. HIV-infected patients on effective anti-retroviral therapy with undetectable viral load within 6 months are eligible for this trial
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Meets other institutional and federal requirements for infectious disease titer requirements

  * Note Infectious disease testing to be performed within 28 days prior to day 1 of protocol therapy
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Women of childbearing potential (WOCBP): Negative urine or serum pregnancy test. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required
* PATIENTS AFTER DAY +30 FOLLOWING CELLULAR THERAPY: Agreement by females and males of childbearing potential\* to use an effective method of birth control or abstain from heterosexual activity for the course of the study through at least 120 days after the last dose of protocol therapy.

  * Childbearing potential defined as not being surgically sterilized (men and women) or have not been free from menses for \> 1 year (women only)

Exclusion Criteria:

* Prior treatment failure with asciminib
* Treatment with strong inducers of CYP3A is not allowed and should be switched to an alternative at least 1 week prior to the start of study treatment
* ARM 1: Treatment with prior HSCT is allowed
* ARM 2: Treatment with prior CAR T cell therapy is allowed
* Cardiac or cardiac repolarization abnormality, including any of the following:

  * History within 6 months prior to starting study treatment of myocardial infarction (MI), or coronary artery bypass graft (CABG)
  * 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)
  * Fridericia's corrected QT interval (QTcF) at screening ≥ 450 msec (male patients), ≥ 470 msec (female patients)
  * Long QT syndrome, family history of idiopathic sudden death or congenital long QT syndrome
* History of pancreatitis within 1 year of study entry or past medical history of chronic pancreatitis
* ARM 1: Active grade 3 or higher graft-versus-host disease (GVHD) after allogeneic HSCT within 14 days of enrollment. Note: prednisone administration (flat dose of 0.5 mg/kg) is allowed. Patients receiving any other medication to control active/progressive GVHD will be excluded
* Clinically significant uncontrolled illness
* Active infection not responding to treatment
* Other active malignancy. Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial
* Females only: Pregnant or breastfeeding
* Any other condition that would, in the Investigator's judgment, contraindicate the patient's participation in the clinical study due to safety concerns with clinical study procedures
* Prospective participants who, in the opinion of the investigator, may not be able to comply with all study procedures (including compliance issues related to feasibility/logistics)