The weaker Vizing conjecture for layer projections contained in minimum dominating sets

Let $G$ and $H$ be graphs, let $G\Box H$ be their Cartesian product, and let $D$ be a dominating set of $G\Box H$. For each $h\in V(H)$, let $G_h$ be the $G$-layer corresponding to $h$, and let $p_G$ denote projection onto $G$. The weaker Vizing conjecture. If, for every $h\in V(H)$, there is a minimum dominating set $D_h$ of $G$ such that

$$p_G(D\cap G_h)\subseteq D_h,$$

then

$$|D|\geq \gamma(G)\gamma(H).$$

This is a special case of Vizing's conjecture, motivated by the paper's discussion of vertically undominated cells and equality in the general half-bound; the paper establishes related special cases but leaves this formulation as the object of conjectural interest.