  We prove many new cases of a conjecture of Calegari-Emerton describing the
qualitative properties of completed cohomology. The heart of our argument is a
careful inductive analysis of completed cohomology on the Borel-Serre boundary.
As a key input to this induction, we prove a new perfectoidness result for
towers of minimally compactified Shimura varieties of pre-abelian type,
generalizing previous work of Scholze.
