Gravier et al.'s chromatic-choosability conjecture for claw-free perfect graphs

A graph is claw-free perfect if it is both claw-free and perfect. Let $\chi(G)$ and $\chi_\ell(G)$ denote its chromatic and list chromatic numbers. Gravier et al.'s conjecture. For every claw-free perfect graph $G$,

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

The conjecture is known for claw-free perfect graphs with clique number at most four, but remains open for non-peculiar claw-free perfect graphs with clique number at least five.