Hochster's finiteness conjecture for rank-one maximal Cohen–Macaulay modules

Let $R$ be a local normal domain over an algebraically closed field. The divisor class group $\operatorname{Cl}(R)$ is the group of rank-one reflexive classes. Hochster's finiteness conjecture. If $\operatorname{Cl}(R)$ is finitely generated, then it contains only finitely many maximal Cohen–Macaulay modules of rank one up to isomorphism. The question is motivated by finiteness problems for Cohen–Macaulay ideals; the paper notes a counterexample to the unrestricted formulation in dimension two and proposes this algebraically closed-field version.