The finite-length Gorenstein conjecture

Let $R$ be a commutative Noetherian local ring and let $M$ be a finitely generated $R$-module. Assume that $M$ has finite length and that $\operatorname{Gdim}(M)$, the Gorenstein dimension of $M$, is finite. Finite-length Gorenstein conjecture. Then $R$ is Gorenstein. The source calls this a well-known conjecture connected with the preceding grade conjecture; no resolution is given in the supplied text.