Gravier–Maffray list-coloring conjecture for claw-free graphs

A graph is claw-free if it has no induced subgraph isomorphic to $K_{1,3}$. Let $\chi(G)$ and $\chi_\ell(G)$ denote the chromatic and list chromatic numbers. Gravier–Maffray's conjecture. For every claw-free graph $G$,

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

This is the broad chromatic-choosability conjecture for claw-free graphs; the supplied text does not resolve it in general.