The Second Brauer-Thrall conjecture for maximal Cohen–Macaulay modules

Let $(R,\mathfrak{m},\mathsf{k})$ be a local ring. Its CM type is unbounded if the multiplicities of its indecomposable maximal Cohen–Macaulay $R$-modules are unbounded, and strongly unbounded if there is an increasing sequence $n_1<n_2<\cdots$ of positive integers such that, for every $i$, infinitely many indecomposable maximal Cohen–Macaulay modules have multiplicity $n_i$.

Second Brauer-Thrall conjecture for MCM modules. If $R$ has unbounded CM type and $\mathsf{k}$ is infinite, then $R$ has strongly unbounded CM type.

The source discusses this as a conjecture and notes that it holds in dimension one in the cases treated there, but the supplied status evidence is mismatched and does not establish a resolution of BTM2.