Existence of generalized quasi t-fine rings outside the generalized fine class

Determine whether, if no quasi t-fine ring is non-fine, there exists a generalized quasi t-fine ring that is not generalized fine.

Background

Generalized quasi t-fine rings are defined by the decomposition R\setminus J(R)=T(R)+Q(R), with Q(R) the set of quasinilpotent elements. The paper establishes examples of generalized quasi t-fine rings that are not generalized t-fine, but it does not determine whether the generalized quasi t-fine class is strictly broader than the generalized fine class.

The concluding problem asks this as a fallback question conditional on the nonexistence of a quasi t-fine ring that is not fine; the authors explicitly identify the answer to the stated problem as unknown.