Noel's extremal complete multipartite graph conjecture

Let $G$ be a $k$-chromatic graph with at most $mk$ vertices, and let $K_{m*k}$ denote the complete $k$-partite graph with $m$ vertices in each part. Noel's conjecture.

$$\chi_\ell(G) \leq \chi_\ell(K_{m*k}).$$

The conjecture asserts that this balanced complete multipartite graph maximizes list chromatic number among the specified $k$-chromatic graphs; the supplied text gives no resolution.