Foxby's conjecture on faithful modules with minimal top Bass number

Let $(A,\mathfrak{m})$ be a Noetherian local ring of dimension $d$, and let $M$ be a finitely generated $A$-module. For a prime ideal $P$ of $A$, write

$$\mu_i(P,M)=\dim_{\kappa(P)}\operatorname{Ext}^i_{A_P}(\kappa(P),M_P),$$

where $\kappa(P)$ is the residue field of $A_P$; this is the $i$th Bass number of $M$ with respect to $P$. A module is faithful if its annihilator is zero, and a canonical module is a canonical module of the local ring $A$.

Foxby's conjecture. If $M$ is faithful and

$$\mu_d(\mathfrak{m},M)=1,$$

then $A$ is Cohen–Macaulay and $M$ is a canonical module of $A$.

For a Cohen–Macaulay ring, a faithful maximal Cohen–Macaulay module with this Bass-number condition is already the canonical module, while the conjecture asks for the conclusion without assuming that $A$ or $M$ is Cohen–Macaulay. The paper presents this as a conjecture attributed essentially to Foxby; the supplied source does not establish its resolution.