The list $\Delta$-Equitable Coloring Conjecture

Let $G$ be a connected 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 $L$. Let $K_m$, $C_{2m+1}$, and $K_{2m+1,2m+1}$ denote respectively a complete graph, an odd cycle, and a complete bipartite graph with parts of size $2m+1$.

List $\Delta$-ECC. $G$ is equitably $k$-choosable for each $k\geq\Delta(G)$ if it is different from $K_m$, $C_{2m+1}$, and $K_{2m+1,2m+1}$.

This is the list-context analogue of the $\Delta$-Equitable Coloring Conjecture. The source presents it as a conjecture attributed to KP03, but gives no evidence of a resolution.