Havet et al.'s list square coloring conjecture for planar graphs

Let $G$ be a planar graph, and let $G^2$ denote its square, in which two vertices are adjacent when their distance in $G$ is at most two. Havet et al.'s conjecture. For any planar graph $G$,

$$\chi_\ell(G^2)=\chi(G^2).$$

This is the planar-graph specialization of list square coloring; the supplied text says the conjecture was later disproved by Hasanvand.