The bipartite list-cover independent transversal conjecture

Let $G$ and $H$ be bipartite graphs, with bipartitions $V(G)=A_G\cup B_G$ and $V(H)=A_H\cup B_H$. Let $H$ be a bipartite list-cover of $G$ with respect to a mapping $L$, meaning that $L$ induces a partition of $V(H)$ aligned with the two bipartitions, and call $H$ a $k$-fold cover when $|L(v)|=k$ for every $v\in V(G)$. An independent transversal is an independent set in $H$ intersecting every part of the partition induced by $L$ exactly once. The bipartite list-cover conjecture. There is some $C\ge1$ such that, for any bipartite graph $G$ of maximum degree $\Delta\ge2$, any

$$\left\lceil C\log_2\Delta\right\rceil$$

-fold bipartite list-cover of $G$ admits an independent transversal. This is a cover-graph reformulation of the logarithmic list-colouring conjecture and remains open.