Existence of quasi t-fine rings outside the fine class

Determine whether there exists a quasi t-fine ring that is not fine.

Background

A quasi t-fine ring is defined as a ring in which every nonzero element is a sum of a torsion unit and a quasinilpotent element; the paper observes that every t-fine ring is quasi t-fine. Example \ref{ex 33}(3) supplies a quasi t-fine ring, but it is also fine, so it does not resolve whether the quasi t-fine class properly extends the fine class.

The authors explicitly state that the existence of a quasi t-fine ring that is not fine is unknown and formulate it as the first alternative in the concluding problem.