The b-chromatic number inequality for powers of Cartesian graph products

Let $G$ and $H$ be graphs, let $k\text{?}$ be a positive integer, and write $\varphi$ for the $b$-chromatic number and $G^k$ for the $k$-th graph power. Cartesian-product power conjecture. For all graphs $G$ and $H$ and all $k\in\mathbb{N}$,

$$\varphi((G\square H)^k)\geq\varphi(G^k\square H^k).$$

The supplied text presents this inequality as a conjecture, and gives no evidence that it has been resolved.