Rall's complete-factor conjecture for well-dominated Cartesian products

Let $G$ and $H$ be nontrivial connected graphs, and let $G\Box H$ denote their Cartesian product. A graph is well-dominated if every minimal dominating set is a minimum dominating set.

Rall's conjecture. If $G\Box H$ is well-dominated, then at least one of $G$ or $H$ is a complete graph.

This conjecture is the explicit final formulation of the proposed classification of connected well-dominated Cartesian products. The paper proves the corresponding characterization when one factor is complete, but the assertion for arbitrary connected factors remains open.