Borodin–Kostochka–Woodall List Total Coloring Conjecture

Let $G$ be a graph, and let $T(G)$ be its total graph, whose vertices are the vertices and edges of $G$, with adjacency given by adjacency, incidence, or edge adjacency in $G$. The source also writes $T(G)=B(G)^2$. Write $\chi$ for chromatic number and $\chi_\ell$ for list chromatic number. Borodin–Kostochka–Woodall List Total Coloring Conjecture. Every graph $G$ satisfies

$$\chi(T(G))=\chi_\ell(T(G)).$$

This conjecture is a special case of the stronger List Square Coloring Conjecture, which has been refuted by Kim and Park; consequently, the total-coloring conjecture is refuted as stated.