Mohar's conjecture on edge-critical graphs and strong chromatic-choosability

Let $G$ be a graph with maximum degree $\Delta(G)$. It is $(\Delta(G)+1)$-edge-critical if it is $(\Delta(G)+1)$-edge-colorable, while every proper subgraph obtained by deleting at least one edge is $\Delta(G)$-edge-colorable. Let $L(G)$ be the line graph of $G$. A graph is strong $k$-chromatic-choosable when it has chromatic number greater than $k-1$ and satisfies the corresponding strong list-coloring condition described in the source.

Mohar's conjecture. If $G$ is a $(\Delta(G)+1)$-edge-critical graph, then $L(G)$ is strong $(\Delta(G)+1)$-chromatic-choosable.

The conjecture was made by Mohar in 2001 and is presented in the source as an unresolved conjecture. The definition of strong chromatic-choosability is specialized, so its precise list-coloring formulation should be checked against the paper.