Kostochka–Pelsmajer–West conjecture on equitable list colorability

Let $G$ be a graph, let $k$ be a positive integer, and let $L$ be a $k$-list assignment for $G$. Write $m:=\left\lceil\frac{|G|}{k}\right\rceil$. An $L$-coloring is $m$-bounded if every color is used on at most $m$ vertices; $G$ is equitably list $k$-colorable (EL $k$-colorable) if it has an $m$-bounded $L$-coloring for every $k$-list assignment $L$. Here $\Delta(G)$ denotes the maximum degree of $G$. Kostochka–Pelsmajer–West conjecture. Every graph $G$ is EL $k$-colorable for each $k>\Delta(G)$. This is the list analogue of equitable colorability and was proposed by Kostochka, Pelsmajer and West. Its resolution is not specified in the supplied text.