Cambie–Kang conjecture for triangle-free DP-coloring

Let $G$ be a graph and let $\mathcal{H}=(L,H)$ be a DP-cover of $G$. A proper $\mathcal{H}$-coloring chooses one color from each $L(v)$ so that the chosen colors are independent in $H$.

Cambie–Kang conjecture. For every $\varepsilon>0$, there is $d_0\in\mathbb{N}$ such that, if $H$ has maximum degree $d\geqslant d_0$ and

$$|L(v)|\geqslant (1+\varepsilon)d/\log d$$

for every $v\in V(G)$, then $G$ admits a proper $\mathcal{H}$-coloring provided either $G$ is triangle-free (weak version) or $H$ is triangle-free (strong version).

The conjecture asks whether the constant $4+o(1)$ can be reduced to $1+o(1)$, and the source explicitly describes it as open, even in the list-coloring framework.