Vasconcelos's finiteness conjecture for semidualizing modules

Let $(R,\mathfrak{m})$ be a local ring. A finitely generated $R$-module $C$ is semidualizing if the homothety morphism

$$R\longrightarrow\operatorname{Hom}_{R}(C,C)$$

is an isomorphism and $\operatorname{Ext}_{R}^{i}(C,C)=0$ for all $i\geqslant 1$. Let $\mathfrak{S}_0(R)$ denote the set of isomorphism classes of semidualizing $R$-modules. Vasconcelos's semidualizing-module conjecture. If $R$ is Cohen–Macaulay, then $\mathfrak{S}_0(R)$ is finite. The supplied text presents this as an open conjecture about the size of the semidualizing-module class.