The list-chromatic-index criticality conjecture

Let $G$ be a graph. Write $\chi'_\ell(G)$ for its list chromatic index and $\chi'(G)$ for its chromatic index. A graph is $\chi'_\ell$-critical if every proper subgraph $H$ satisfies $\chi'_\ell(H)<\chi'_\ell(G)$; analogously, it is $\chi'$-critical if every proper subgraph $H$ satisfies $\chi'(H)<\chi'(G)$.

List-chromatic-index criticality conjecture. Every $\chi'_\ell$-critical graph is $\chi'$-critical.

The conjecture is presented as a possible approach to the List Coloring Conjecture, but no resolution is given in the supplied source.