The dimension conjecture for Cohen–Macaulay categories of countable representation type

Let $R$ be a Cohen–Macaulay local ring. Write $\operatorname{CM}(R)$ for the category of maximal Cohen–Macaulay $R$-modules, and let $\dim\operatorname{CM}(R)$ denote its dimension.

Dimension conjecture. If $R$ has countable $\operatorname{CM}$-representation type, then

$$\dim\operatorname{CM}(R)\le 1.$$

The conjecture is known when $R$ has finite $\operatorname{CM}$-representation type. The paper relates it to the dimension of the stable category in the Gorenstein case, but does not settle it in general.