Weighted Graham-type conjecture for cover pebbling numbers

Let $G$ and $H$ be graphs. A nonnegative function on a graph assigns a nonnegative real number to each vertex. Let $\omega_1$ be a nonnegative function on $G$ and $\omega_2$ a nonnegative function on $H$, and define $\omega$ on the Cartesian product $G\times H$ by

$$\omega((g,h))=\omega_1(g)\omega_2(h),\qquad g\in V(G),\ h\in V(H).$$

The weighted cover pebbling number $\gamma_\omega(G)$ is the minimum number of pebbles sufficient to meet the vertex demands specified by $\omega$, regardless of the initial distribution. Weighted Graham-type conjecture. The weighted cover pebbling number satisfies

$$\gamma_\omega(G\times H)\leq \gamma_{\omega_1}(G)\gamma_{\omega_2}(H).$$

This is proposed as a conjecture analogous to Graham's conjecture. The paper presents it after stating related open problems, and no resolution is supplied.