The Huneke–Wiegand conjecture for Gorenstein quasi-fiber product rings

Let $R$ be a Gorenstein quasi-fiber product ring. Say that $R$ satisfies (HW) when every torsion-free $R$-module $M$ with rank such that $M\otimes_R M^*$ is maximal Cohen–Macaulay is free, where $M^*=\operatorname{Hom}_R(M,R)$.

Huneke–Wiegand quasi-fiber-product conjecture. Gorenstein quasi-fiber product rings satisfy (HW).

The paper studies this conjecture using vanishing of Ext and proves the corresponding statement for one-dimensional Gorenstein quasi-fiber product rings. The general assertion remains open in the supplied text.