Minimum domination conjecture for the augmented Cartesian product

Let $G$ and $H$ be graphs, let $D_G$ and $D_H$ be minimum dominating sets of $G$ and $H$, respectively, and let $A_{G,H}[F_G,F_H]$ denote the graph obtained by the paper's edge-adjoining construction from the edge sets $F_G$ and $F_H$. The Cartesian product $D_G\times D_H$ is a dominating set in this augmented graph. Minimum domination conjecture. The set $D_G\times D_H$ is a minimum dominating set for

$$A_{G,H}[F_G,F_H].$$

The conjecture is the paper's proposed strengthening of the construction's minimality conclusion: it asserts that the dominating set of cardinality $\gamma(G)\gamma(H)$ is minimum, not merely minimal. The paper states that this conjecture would imply Vizing's conjecture.