Inclusion Criteria:  Individuals who meet all of the following criteria are eligible for enrollment as study participants:  Participant must be able to understand and provide informed consent and be willing to comply with study procedures and follow up; Meet the American College of Rheumatology/European League Against Rheumatism (ACR/EULAR) Classification Criteria for IgG4-Related Disease (IgG4-RD); Have active disease based at screening on an IgG4-RD Responder Index (RI) ≥4, with disease manifestations in at least two organ systems;  May have newly-diagnosed or relapsing disease at screening  --Relapsing disease is defined as IgG4-RD that has previously been in remission but is now active again;  May be on treatment or off treatment for IgG4-RD at the time of screening  --If on treatment, must be willing to discontinue those other treatments before the Baseline (Day 0) visit;  No history of severe allergic reactions to monoclonal antibodies; Female participants of childbearing potential must have a negative pregnancy test upon study entry; Female participants of childbearing potential and male participants with a partner of childbearing potential must agree to use Food and Drug Administration (FDA) approved methods of birth control for the entire duration of the study; and,  Immunization with one of the FDA authorized or licensed Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2) vaccines is required for study entry  Vaccination series must have been completed at least 2 weeks prior to start of study therapy Participants with Coronavirus Disease 2019 (COVID-19) infections within the preceding three months must have 2 consecutive negative nasal swab Polymerase Chain Reaction (PCR) tests performed at least 24 hours apart.  Exclusion Criteria:  Individuals who meet any of these criteria are not eligible for enrollment as study participants:  Treatment with more than 40 mg/day of prednisone within 28 days of the Baseline (Day 0) visit; Presence of a condition other than IgG4-RD that (e.g., asthma) is likely to require systemic glucocorticoids (GC) for disease control during the period of the trial; Malignancy within 5 years (except successfully treated in situ cervical cancer, resected squamous cell or basal cell carcinoma of the skin);  The following lab values as indicators of hepatic dysfunction:  Serum aspartate aminotransferase (AST) or alanine aminotransferase (ALT) greater (>) than three times the upper limit of normal (ULN)  Total bilirubin >two times the ULN unless caused by Gilbert's disease  ---Gilbert's disease with total bilirubin > three times ULN  Serum albumin <2.5 mg/dL; Evidence of another uncontrolled condition which, in the judgment of the investigator, could interfere with participation in the trial according to the protocol; Active infection requiring hospitalization or treatment with systemic antimicrobial agents within the 30 days prior to randomization; Prior use of rituximab or other B cell depleting agents within 9 months of enrollment, unless B cells have been demonstrated to have repopulated; Use of any investigational agent or biologic and non-biologic disease-modifying antirheumatic drugs (DMARDs) within 5 half-lives of the agent (or 6 months if the half- life is unknown) prior to enrollment;  Any of the following laboratory tests at the Screening Visit:  White blood cell count (WBC) <3.0 x 10^3/microliter (µL) Absolute neutrophil count (ANC) <1.5 x 10^3/µL Hemoglobin <10 g/dL Platelet count <75 x 10^9/L Estimated glomerular filtration rate (eGFR) ≤45 ml/minute/1.73m^2; The use of supplemental oxygen at baseline (Day 0);  Positive Quantiferon gold assay  Indeterminate Quantiferon gold assays must be repeated (with same or other interferon gamma release assay (IGRA) per local policy) and shown to be negative  ---Alternatively, if the Quantiferon gold assay remains indeterminant, a participant must have a negative purified protein derivative (PPD)  If the participant has had the Bacillus Calmette-Guérin (BCG) vaccine or has some other condition complicating the interpretation of tuberculosis (TB) testing, consultation with infectious disease specialist must be obtained before receipt of the first investigational infusion ----Note: Participants diagnosed with latent TB are eligible but must have received appropriate prophylaxis for 30 days before their first investigational infusion;  Medical history or serologic evidence at Screening of chronic infections including:  Human immunodeficiency virus (HIV) infection Hepatitis B as indicated by surface antigen or hepatitis B core antibody positivity Hepatitis C as indicated by anti-hepatitis C antibody positivity ---If a participant is Hepatitis C antibody positive, they will be eligible to participate in the study if he/she is negative for viral load at Screening; Live vaccines within 8 weeks of initiating study therapy; Participant is pregnant or breastfeeding, or planning a pregnancy while enrolled in the study; Substance use disorder, including the recurrent use of alcohol and/or drugs within the past year associated with clinically significant impairment associated with failure to meet major responsibilities at work, school, or home;  IgG4-RD that is dominated primarily by advanced fibrotic lesions (°)  --Specifically, participants whose disease manifestations consist only of:  retroperitoneal fibrosis, fibrosing mediastinitis, sclerosing mesenteritis, or Riedel's thyroiditis  (°) Participants with these (Exclusion item 16) disease manifestations can be included, however, only if they have disease in 2 organ systems that is not of an advanced fibrotic nature and otherwise meet the Inclusion and Exclusion Criteria; or,  Co-enrollment: While participating in AIG01, participants may not be in another interventional trial, but may be in observational registries or cohorts as long as the total combined volume of blood to be drawn does not exceed the National Institutes of Health (NIH) limit of and objectives do not confound the current study.