Havet–van den Heuvel–McDiarmid–Reed conjecture for nice graph classes

A graph class $\mathcal{G}$ is nice if it is minor-closed and does not contain $K_{3,t}$ for some positive integer $t$. Havet–van den Heuvel–McDiarmid–Reed conjecture. There exists $\Delta_0$ such that, whenever $G\in\mathcal{G}$ and $\Delta(G)\ge\Delta_0$,

$$\chi^2(G)\le\chi_{\ell}^2(G)\le\left\lfloor\frac{3}{2}\Delta(G)\right\rfloor+1.$$

This asymptotic strengthening is posed for all nice classes; the survey gives asymptotic results for list coloring but does not report a resolution of this sharper bound.