Schoutens's conjecture on very small maximal Cohen–Macaulay modules

Let $(R,\mathfrak{m})$ be a complete local ring in equal characteristic. A finitely generated maximal Cohen–Macaulay $R$-module $M$ is very small if

$$e(M)\leq\min\{l(R/I)\mid I\text{ is generated by a system of parameters}\},$$

where $e(M)$ is the multiplicity of $M$ with respect to $\mathfrak{m}$.

Schoutens's conjecture. Every complete local ring admits a very small maximal Cohen–Macaulay module.

Very small maximal Cohen–Macaulay modules strengthen the existence of small maximal Cohen–Macaulay modules. The source describes existence results in dimension at most two and for affine toric rings, but provides no resolution of the conjecture.