Does finite Ext gap for a module imply a finite Auslander bound?

Determine whether, for a finitely generated module M over a Cohen–Macaulay local ring R, the finiteness of the Ext gap Extgap_R(M)—defined by Extgap_R(M) = sup_N d_R(M,N) over all finitely generated R-modules N, where d_R(M,N) is the largest r such that there exists n with Ext_R^{n+1}(M,N) = ... = Ext_R^{n+r}(M,N) = 0 but Ext_R^{n}(M,N) ≠ 0 and Ext_R^{n+r+1}(M,N) ≠ 0—implies that the Auslander bound b_M = sup{ PR(M,N) | PR(M,N) < ∞ and N finitely generated } is finite, where PR(M,N) = sup{i | Ext_R^i(M,N) ≠ 0}.

Background

The paper studies Auslander bounds b_M of modules, generalizing projective dimension, and explores conditions ensuring finiteness of b_M. Section 3 introduces the notion of Ext gaps (Extgap_R(M)) and Ext-bounded rings, proving that if R is Ext bounded and PR(M,R) is finite, then b_M and Gdim_R M are finite (Corollary 3.13), extending results of Huneke–Jorgensen.

Motivated by moving from ring-wide assumptions to module-specific ones, the authors ask whether merely having a finite Ext gap for a single module M (a weaker hypothesis than R being Ext bounded) suffices to guarantee that M has finite Auslander bound.