List Total Coloring Conjecture

For a multigraph $G$, a total coloring colors vertices and edges so that adjacent vertices, incident edges, and incident vertex-edge pairs receive distinct colors. Let $\chi”(G)$ be the total chromatic number and $\chi”_\ell(G)$ the list total chromatic number. List Total Coloring Conjecture. For any multigraph $G$,

$$\chi”_\ell(G)=\chi”(G).$$

The conjecture is known for several classes, including multigraphs whose underlying simple graph is a forest or circuit and bipartite graphs with total chromatic number $\Delta+2$; no general resolution is given.