The study hypothesis is that the administration of posoleucel (ALVR105) to patients with virus-associated HC will demonstrate superiority for the time to resolution of HC (as measured by resolution of macroscopic hematuria) compared to patients treated with placebo. The primary hypothesis will be tested in patients with BK virus (BKV) viruria to demonstrate superiority over placebo in this population (BK Intent-to-Treat \[ITT\] Population). A supplementary analysis will be conducted in all patients with any virus-associated HC (cytomegalovirus \[CMV\], human herpesvirus 6 \[HHV-6\], Epstein-Barr virus \[EBV\], JC virus \[JCV\], and/or adenovirus \[AdV\]) in order to evaluate efficacy in this broader population (ITT Population).