The strong-product infinity conjecture for distance packing domination

Let $G$ and $H$ be graphs, and let $d$ and $p$ be integers. The $d$-distance $p$-packing domination number $\gamma_d^p(G)$ is the cardinality of a smallest set of vertices that is both a $d$-distance dominating set and a $p$-packing; if no such set exists, it is defined to be infinity. The strong product $G\boxtimes H$ has vertex set $V(G)\times V(H)$, with adjacency inherited from adjacency in either factor or both factors.

Strong-product infinity conjecture. If

$$\gamma_d^p(G)=\infty,$$

then

$$\gamma_d^p(G\boxtimes H)=\infty$$

for every graph $H$.

The conjecture asks whether the nonexistence of a $d$-distance $p$-packing dominating set in one factor necessarily persists after taking its strong product with any graph. The paper establishes the upper bound $\gamma_d^p(G\boxtimes H)\leq \gamma_d^p(G)\gamma_d^p(H)$ when the relevant quantities are finite, but the converse implication concerning infinity is left as a conjecture.