Conjecture on unbounded ranks of small maximal Cohen–Macaulay modules

A small maximal Cohen–Macaulay module over a local ring is a finitely generated maximal Cohen–Macaulay module whose multiplicity satisfies the smallness bound discussed in the source. For a positive integer $N$, saying that a ring admits no small maximal Cohen–Macaulay module of rank at most $N$ means that every such module has rank greater than $N$.

Unbounded-rank small maximal Cohen–Macaulay conjecture. For every integer $N$, there exists a complete local domain $R$ that does not admit small maximal Cohen–Macaulay modules of rank $\leq N$.

This conjecture is motivated by examples where small maximal Cohen–Macaulay modules exist only in higher rank, while the preceding result rules out rank-one maximal Cohen–Macaulay modules in a family of complete local domains. The source gives no resolution.