Finiteness conjecture for maximal Cohen–Macaulay elements in the class group

Let $R$ be a complete local Cohen–Macaulay ring satisfying Serre's condition $(R_2)$, meaning that $R_{\mathfrak p}$ is regular for every prime $\mathfrak p$ of height at most two. The class group of $R$ consists of rank-one reflexive modules modulo the usual class-group equivalence; an element is MCM when its representative is a maximal Cohen–Macaulay module. Class-group finiteness conjecture. The set of MCM elements in the class group of $R$ is finite. This is presented as a link between the finiteness conjectures for rigid MCM modules and modifying modules, whose endomorphism rings are MCM; its general status is open.