Takahashi's finite-length module conjecture for Cohen–Macaulayness

Let $R$ be a Noetherian local ring. Suppose that there exists an $R$-module $M$ of finite length and finite G-dimension.

Takahashi's conjecture. Then $R$ is Cohen–Macaulay.

This conjecture asks whether the existence of a finite-length module of finite G-dimension forces the ring to be Cohen–Macaulay. Its status is not determined by the supplied source context.