Three-set extension of Vizing's conjecture for Cartesian products

Let $G$ and $H$ be graphs, let $G\Box H$ be their Cartesian product, and let $D$ be a dominating set of $G\Box H$. Let $M_G(A)$ denote the family of dominating sets of $G$ associated with the projection $A$ as defined in the source. Three-set Vizing conjecture. If there exist dominating sets $S_1,S_2,S_3$ of $G$ such that, for every $h\in V(H)$, one has

$$S_i\in M_G\bigl(p_G(D\cap G_h)\bigr)$$

for some $i\in[3]$, then

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

The source describes this as a step beyond its proved results and as a special case of Vizing's conjecture. It generalizes the preceding two-set situation, but the notation $M_G$ is only used through its defining role in the paper.