Asymptotic independent-transversal packing bounds for list and correspondence covers

For each $D$, let $\Lambda^\star_\ell(D)$ and $\Lambda^\star_c(D)$ be the least fold numbers guaranteeing an independent-transversal packing for, respectively, list-covers and correspondence-covers, under the condition that the cover graph has maximum degree at most $D$.

Independent-transversal packing conjecture.

$$\Lambda^\star_\ell(D) \le D+o(D) \quad\text{and}\quad \Lambda^\star_c(D) \le D+o(D)$$

as $D\to\infty$.

The source records the general upper bound $\Lambda^\star_\ell(D)\le\Lambda^\star_c(D)\le 2D+o(D)$, so the conjecture asks for an asymptotic improvement to leading constant $1$. It remains open.