Dhawan's multipartite hypergraph choosability conjecture

Let $k\geqslant 2$. A $k$-partite $k$-graph is a $k$-uniform hypergraph whose vertex set has a partition into $k$ parts such that every edge contains exactly one vertex from each part. Let $H$ be such a hypergraph with maximum degree at most $\Delta$, and let $\mathsf{ch}(H)$ denote its choice number.

Dhawan's conjecture. For every $k\geqslant 2$, there is a constant $c=c(k)>0$ such that, for all sufficiently large $\Delta$,

$$\mathsf{ch}(H)\leqslant c\log \Delta.$$

This is a stronger uniform statement extending the bipartite-graph conjecture to all uniformities. A recent bound of order $(\Delta/\log\Delta)^{1/(k-1)}$ is known, but the conjectured logarithmic bound remains unresolved.