Kostochka–Woodall conjecture on the choice number of graph squares

Let $G$ be a simple graph. Its square $G^2$ is the graph on $V(G)$ in which two vertices are adjacent when their distance in $G$ is at most two. Let $\chi(G^2)$ denote its chromatic number and let $\chi_l(G^2)$ denote its choice number, the least $k$ such that every assignment of lists of $k$ colors to the vertices admits a proper list-coloring.

Kostochka–Woodall conjecture.

$$\chi_l(G^2)=\chi(G^2).$$

This asserts that every graph square is chromatic-choosable. The conjecture is presented as a motivation for the paper's list-coloring results, but no resolution is given in the supplied text.