Cohen–Macaulay descent under tensoring with a perfect module

Let $(R,\mathfrak m)$ be a Cohen–Macaulay local ring, let $M$ be a perfect $R$-module of positive dimension, and let $N$ be an $R$-module such that

$$\dim(N)=\dim(R).$$

Cohen–Macaulay descent conjecture. If $M\otimes_R N$ is a Cohen–Macaulay $R$-module, then $N$ is a maximal Cohen–Macaulay $R$-module. This conjecture asks whether Cohen–Macaulayness of the tensor product descends to the full-dimensional factor under these hypotheses; its resolution is not specified in the source.