List chromatic number of bipartite graphs

Prove or disprove that every bipartite graph of maximum degree Δ has list chromatic number O(log Δ).

Background

The paper observes that the local-occupancy methods developed for triangle-free graphs do not exploit the global bipartition of a bipartite graph. Although those methods yield a general triangle-free bound of order Δ/log Δ, Alon and Krivelevich conjectured that the much stronger order-logarithmic bound holds for bipartite graphs. The paper states that this conjecture remains unresolved despite progress by non-local methods.

References

Alon and Krivelevich have boldly conjectured that the answer to the above question is at most $O(\log \Delta)$ -- a quantity that is dramatically smaller than $\Theta(\Delta/\log\Delta)$ which we have in the triangle-free case.

— The hard-core model in graph theory  (2501.03379 - Davies et al., 6 Jan 2025) in Section “Barriers”