Generalized Auslander–Reiten conjecture for local rings

Let $R$ be the local ring under consideration and let $M$ be an $R$-module. Generalized Auslander–Reiten conjecture. If

$$\operatorname{Ext}^n_R(M,M\oplus R)=0$$

for all $n\gg 1$, then $M$ has finite projective dimension. This conjecture generalizes the Auslander–Reiten conjecture by requiring Ext-vanishing only in sufficiently large degrees. An instance where the Auslander–Reiten conjecture is known to hold but the generalized conjecture remains open is given by Cohen–Macaulay normal rings.