Alon's logarithmic bound conjecture for bipartite graph list coloring

Let $G$ be a bipartite graph, and let $\Delta(G)$ denote its maximum degree. Alon's conjecture. There exists a constant $c$ such that

$$\chi_\ell(G) \leq c \log \Delta(G).$$

This conjecture concerns whether bipartite graphs have list chromatic number only logarithmic in maximum degree; the supplied text presents it as an open problem.