Cranston–Kim conjecture on list coloring squares of graphs

Let $G$ be a connected graph with maximum degree $\Delta\ge 3$. Let $G^2$ be its square, and let $\chi_{\ell}(G^2)$ denote the list chromatic number of $G^2$. A Moore graph is a $\Delta$-regular graph on $\Delta^2+1$ vertices such that $G^2=K_{\Delta^2+1}$.

Cranston–Kim conjecture. If $G$ is not a Moore graph, then

$$\chi_{\ell}(G^2)\le \Delta^2-1.$$

The source states that Cranston and Kim proved the conjecture for $\Delta=3$, while Bonamy and Bousquet proved the full claim in the more general setting of online list coloring.