The list coloring conjecture for edge and total coloring

Let $G$ be a simple graph. Write $\chi'(G)$ and $\chi”(G)$ for its edge-chromatic and total-chromatic numbers, and $\chi'_{\ell}(G)$ and $\chi”_{\ell}(G)$ for the corresponding list-chromatic parameters. List coloring conjectures. Every simple graph $G$ satisfies

$$\chi'(G)=\chi'_{\ell}(G)\quad\text{and}\quad \chi”(G)=\chi”_{\ell}(G).$$

List coloring strengthens ordinary coloring, and equality would show that edge and total coloring have no gap between their ordinary and list versions; the source presents this as an open conjecture.