The necessity of having no isolatable vertex for well-covered prisms

Let $G$ be a connected, triangle-free graph that contains a cycle of order $4$. The prism $G \Box K_2$ is the Cartesian product of $G$ with $K_2$, and a graph is well-covered if all its maximal independent sets have the same cardinality. An isolatable vertex is a vertex that belongs to an independent set whose removal leaves a graph with a unique perfect matching.

Prism necessity conjecture. If $G \Box K_2$ is well-covered, then $G$ has no isolatable vertex.

This conjecture proposes that, for connected triangle-free graphs containing a $4$-cycle, having no isolatable vertex is not only sufficient but also necessary for the prism to be well-covered. The supplied text does not indicate whether the conjecture has been resolved.