Strong-product variation of Graham's pebbling conjecture

Let $G$ and $H$ be connected graphs, let $G\boxtimes H$ denote their strong product, and let $\pi(G)$ denote the pebbling number of a graph. The notation $|G|$ and $|H|$ denotes the numbers of vertices of $G$ and $H$, respectively.

Strong-product pebbling conjecture. For any connected graphs $G$ and $H$,

$$\pi(G\boxtimes H)\leq \max\left\{\frac{1}{2}\pi(G)\pi(H),\ |G||H|\right\}+2.$$

This is proposed as a variation of Graham's conjecture for strong products. The paper gives motivating bounds and an example showing that a simpler factor-$\frac12$ bound does not hold, while this sharper variation remains unresolved in the source.