Krivelevich–Alon conjecture on the list chromatic number of bipartite graphs

Let $G$ be a bipartite graph with maximum degree at most $\Delta$, and let $k$ be a positive integer. The graph is $k$-choosable if every assignment of lists of $k$ colours to its vertices admits a proper colouring from those lists. Krivelevich–Alon conjecture. There is an absolute constant $C>0$ such that $G$ is $k$-choosable whenever $k\geq C\log\Delta$. This conjecture extends the asymptotic behaviour of complete bipartite graphs to all bipartite graphs of maximum degree $\Delta$. The source presents it as an open conjecture and notes that the paper's results give partial progress, especially in asymmetric cases.