Borowiecki–Jozef Cartesian product list-coloring conjecture

For graphs $G_1$ and $G_2$, their Cartesian product $G_1\Box G_2$ has vertex set $V(G_1)\times V(G_2)$, with $(u,v)$ adjacent to $(u',v')$ when either $u=u'$ and $vv'\in E(G_2)$ or $v=v'$ and $uu'\in E(G_1)$. Borowiecki–Jozef's conjecture. For every pair of graphs $G_1$ and $G_2$, there exists a constant $C$ such that

$$\chi_\ell(G_1\Box G_2)\leq C\bigl(\chi_\ell(G_1)+\chi_\ell(G_2)\bigr).$$

The paper presents this as one of two conjectures on Cartesian products and gives no resolution.