The strong product conjecture for Grundy domination

Let $G$ and $H$ be graphs. Their strong product $G\boxtimes H$ has vertex set $V(G)\times V(H)$, with two vertices $(g_1,h_1)$ and $(g_2,h_2)$ adjacent when either $g_1=g_2$ and $h_1$ is adjacent to $h_2$ in $H$, $g_1$ is adjacent to $g_2$ in $G$ and $h_1=h_2$, or $g_1$ is adjacent to $g_2$ in $G$ and $h_1$ is adjacent to $h_2$ in $H$. The parameter $\gamma_{gr}(G)$ denotes the Grundy domination number of $G$. Strong product conjecture. For any graphs $G$ and $H$,

$$\gamma_{gr}(G \boxtimes H) = \gamma_{gr}(G)\gamma_{gr}(H).$$

The inequality $\gamma_{gr}(G \boxtimes H)\geq\gamma_{gr}(G)\gamma_{gr}(H)$ is known, so the conjecture asserts the reverse inequality and would determine the Grundy domination number of every strong product from those of its factors. Its status is not resolved in the supplied source.