The list coloring conjecture for line graphs

Let $G$ be a graph. Its choice number $ch(G)$ is the least integer $k$ such that every assignment of a $k$-element list of colors to each vertex admits a proper coloring from the assigned lists. Its list chromatic index is $ch'(G)=ch(L(G))$, where $L(G)$ is the line graph of $G$, and let $\chi'(G)$ be the chromatic index. List coloring conjecture. For every graph $G$,

$$ch'(G)=\chi'(G).$$

The conjecture says that line graphs have no gap between list chromatic index and chromatic index; no resolution status is supplied in the source span.