Gorensteinness from TM = M without finite G-dimension

Determine whether every commutative noetherian local ring R is Gorenstein whenever there exists a finitely generated R-module M with dim M = dim R such that a characteristic module TM exists and equals M, where TM is defined as Tor_{dim Q − depth R}^Q(R, M) with respect to a Cohen presentation Q → R.

Background

The paper introduces characteristic and cocharacteristic modules for R-modules over a local ring R via higher Tor and Ext with respect to a Cohen presentation Q → R, and uses these to characterize Cohen–Macaulayness and Gorensteinness of R.

Theorem 1.4 establishes several equivalent conditions for R to be Gorenstein, including the existence of an R-module M of full dimension with finite Gorenstein dimension such that the characteristic module TM exists and equals M. The authors then ask whether this finite Gorenstein dimension hypothesis can be removed, i.e., whether TM = M alone forces R to be Gorenstein. They note the question is affirmative in the artinian case by Corollary 3.8.