Revised square-graph chromatic-choosability conjecture

Let $G$ be a square graph, meaning a graph of the form $H^2$ for some graph $H$, and let $\Delta(G)$ denote its maximum degree. Write $\chi$ for chromatic number and $\chi_\ell$ for list chromatic number. Revised square-graph conjecture. Every square graph $G$ satisfying

$$\chi(G)\geq \frac{1}{2}\Delta(G)+1$$

is chromatic-choosable, that is, satisfies $\chi_\ell(G)=\chi(G)$. The source proposes this as a revision after its examples violate the corresponding property; its resolution status is not supplied.