The Gorenstein-dimension grade conjecture

Let $R$ be a commutative Noetherian local ring and let $M$ be a finitely generated $R$-module. Write $\operatorname{Gdim}(M)$ for the Gorenstein dimension, $\operatorname{grade}(M)$ for the grade, and $\operatorname{Ann}(M)$ for the annihilator. Gorenstein-dimension grade conjecture. If

$$\operatorname{Gdim}(M)<\infty,$$

then

$$\operatorname{grade}(M)=\operatorname{ht}(\operatorname{Ann}(M)).$$

The paper proves this in several special cases, including modules of the form $R/P$, but leaves the general statement as a conjecture.