Borowiecki–Jozef asymptotic Cartesian product conjecture

Let $G_1$ and $G_2$ be graphs, and let $\Delta$ be a common upper bound for their maximum degrees, so that $\Delta(G_1)\leq\Delta$ and $\Delta(G_2)\leq\Delta$. Borowiecki–Jozef's asymptotic conjecture.

$$\chi_\ell(G_1\Box G_2)\leq\Delta+o(\Delta).$$

This is the second Cartesian-product conjecture stated in the supplied text; its resolution is not given.