Strengthened Chen–Lih–Wu and Kostochka–Pelsmajer–West conjecture

Let $G$ be an $r$-colorable graph, with maximum degree $\Delta(G)\leq r$. Let an $r$-list assignment $L$ assign $r$ available colors to each vertex, and call $G$ equitably $r$-choosable if every such assignment has a proper list coloring using each color at most $\left\lceil |V(G)|/r\right\rceil$ times. A coloring is SE $r$-choosable when it satisfies the stronger size condition defined in the paper: all color classes have size at most $\left\lceil |V(G)|/r\right\rceil$, with at most $|V(G)|\mathbin{\operatorname{mod}^*}r$ classes attaining that upper bound. Strengthened conjecture. Either $r$ is odd and $K_{r,r}\subseteq G$, or $G$ is equitably $r$-choosable and even SE $r$-choosable. The statement is proposed in the concluding remarks as a possible strengthening; it is not claimed to be proved, and its status is open.