The probabilistic Graham product conjecture

For graphs $G_1$ and $G_2$, let $P_{1/2}(G)$ denote the smallest number of pebbles in a randomly generated configuration that is solvable with probability at least $1/2$. Probabilistic Graham conjecture. There is a universal constant $C$ such that

$$P_{1/2}(G_1\times G_2)\leq C P_{1/2}(G_1)\cdot P_{1/2}(G_2).$$

This is a probabilistic analogue of Graham's Conjecture for pebbling, which predicts a multiplicative bound for the pebbling number of Cartesian products. The conjecture is refuted by a counterexample involving the sequence of hypercubes $Q_i$.