List-colouring extension of Wegner's conjecture for nice graph families

Let $\mathcal{F}$ be a nice family of graphs, and let $G\in\mathcal{F}$ have maximum degree $\Delta(G)$ sufficiently large.

List-colouring extension of Wegner's conjecture.

$$\chi(G^2)\le\mathit{ch}(G^2)\le\left\lfloor\frac32\,\Delta(G)\right\rfloor+1.$$

The source notes that this is true for the family of $K_4$-minor-free graphs, by results of Lih, Wang and Zhu and of Hetherington and Woodall. The conjecture remains open for general nice families in the source.