Häggkvist–Chetwynd line graph list-coloring conjecture

For a graph $G$, its line graph $L(G)$ has one vertex for each edge of $G$, with two vertices adjacent exactly when the corresponding edges share an endpoint. Let $\chi$ and $\chi_\ell$ denote the chromatic and list chromatic numbers. Häggkvist–Chetwynd's conjecture. For any graph $G$,

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

This is the line-graph formulation of the List Edge Coloring Conjecture; the supplied text gives no general resolution.