The list Hajnál–Szemerédi conjecture

Let $G$ be a finite simple graph. A list assignment $L$ assigns a list of colors to each vertex; it is a $k$-assignment when $|L(v)|=k$ for every $v\in V(G)$. An equitable $L$-coloring is a proper coloring using colors from the lists such that each color appears on at most $\lceil |V(G)|/k\rceil$ vertices. The graph is equitably $k$-choosable when it has an equitable $L$-coloring for every $k$-assignment $L$.

List Hajnál–Szemerédi conjecture. Every graph $G$ is equitably $k$-choosable when $k\geq \Delta(G)+1$.

This is the list-coloring analogue of the Hajnál–Szemerédi theorem, which asserts equitable $k$-colorability for $k\geq\Delta(G)+1$. The source attributes the conjecture to Kostochka, West, and the third author; its resolution status is not specified in the supplied text.