Generalized Graham pebbling product conjecture

Let $G_1$ and $G_2$ be graphs, let $p_1,p_2$ be positive integers, and let $f_{(p_1,p_2)}$ denote the corresponding generalized pebbling number, with $f_{p_i}(G_i)$ the one-factor generalized pebbling numbers. Generalized Graham conjecture. For all $G_1,G_2,p_1,p_2$,

$$f_{(p_1,p_2)}(G_1\Box G_2)\le f_{p_1}(G_1)f_{p_2}(G_2).$$

This is presented as a natural generalization of Graham's conjecture; its resolution is not specified in the supplied text.