Asymmetric list-colouring conjecture for bipartite graphs

Let $G=(V=A\cup B,E)$ be a bipartite graph with maximum degrees on $A$ and $B$ at most $\Delta_A$ and $\Delta_B$, respectively. For positive integers $k_A$ and $k_B$, say that $G$ is $(k_A,k_B)$-choosable if every assignment of lists of size $k_A$ to vertices of $A$ and size $k_B$ to vertices of $B$ admits a proper list colouring. Asymmetric list-colouring conjecture. Let the positive integers $\Delta_A$, $\Delta_B$, $k_A$, and $k_B$ satisfy one of the following: given $\varepsilon>0$, $\Delta_A,\Delta_B\geq\Delta_0$ for some $\Delta_0=\Delta_0(\varepsilon)$, with $k_A\geq\Delta_A^\varepsilon$ and $k_B\geq\Delta_B^\varepsilon$; for some absolute constant $C>1$, $k_A\geq C\log\Delta_B$ and $k_B\geq C\log\Delta_A$; or $\Delta_A=\Delta_B=\Delta$ and, for some absolute constant $C>0$, either $k_B\geq C(\Delta/\log\Delta)^{1/k_A}\log\Delta$ or $k_A\geq C(\Delta/\log\Delta)^{1/k_B}\log\Delta$. Then every such graph $G$ is $(k_A,k_B)$-choosable. The paper develops sufficient and necessary conditions for asymmetric list colouring and establishes several asymptotically sharp results, but this full conjecture remains open.