A bipartite-subgraph-free bound for pebbling numbers of direct graph products

Let $G$ and $H$ be connected graphs, with $G\ne K_1$ and $H$ nonbipartite. Let $G\times H$ denote the direct product of graphs, and let $\pi(G)$ denote the pebbling number.

Direct-product pebbling conjecture.

$$\pi(G\times H)\leq \frac{9}{16}\pi(G)\pi(H)^2.$$

The conjecture seeks to remove the use of chosen connected spanning bipartite subgraphs from the preceding results for direct products. The paper presents the inequality as an expected analogous bound, and it remains unresolved in the source.