Hochster's small Cohen–Macaulay conjecture

Let $(R,\mathfrak{m})$ be a complete local ring. A small Cohen–Macaulay module over $R$ is a nonzero finitely generated $R$-module $S$ such that $\operatorname{depth}(S)=\dim(R)$. Hochster's small Cohen–Macaulay conjecture. Every complete local ring admits a small Cohen–Macaulay module. The conjecture asks whether finite Cohen–Macaulay modules exist for all complete local rings; the paper's abstract indicates that, in mixed characteristic and dimension at least four, there are normal local domains with no small Cohen–Macaulay algebra, while still admitting small Cohen–Macaulay modules.