Vizing–Gupta–Albertson–Collins–Bollobás–Harris line-graph List Coloring Conjecture

Let $G$ be a graph and let $L(G)$ be its line graph, whose vertices are the edges of $G$, with two vertices adjacent when the corresponding edges share an endpoint. Write $\chi$ for chromatic number and $\chi_\ell$ for list chromatic number. Line-graph List Coloring Conjecture. Every graph $G$ satisfies

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

The source notes that this conjecture is proved for bipartite graphs, while its general status is not resolved in the supplied material.