DP-coloring extension of the Alon–Krivelevich–Sudakov conjecture

Let $G$ be a graph and let $\mathcal{H}=(L,H)$ be a DP-cover of $G$. A proper $\mathcal{H}$-coloring selects one color from each $L(u)$ so that the selected colors form an independent set in $H$.

DP-coloring extension of the Alon–Krivelevich–Sudakov conjecture. For every graph $F$, there is a constant $c_F>0$ such that, if $H$ is $F$-free, has maximum degree $d\geqslant 2$, and

$$|L(u)|\geqslant c_Fd/\log d$$

for every $u\in V(G)$, then $G$ admits a proper $\mathcal{H}$-coloring.

This would strengthen the ordinary coloring conjecture by imposing the forbidden-subgraph condition on the DP-cover graph rather than on the base graph. The source presents this stronger form as a conjectural extension, and no resolution is given.