Eventual equality of list colorings and colorings for edgeless graphs

Let $\overline{K}_n$ be an unlabeled, edgeless graph on $n$ vertices. The functions $P_{\ell}(\overline{K}_n,k)$ and $P(\overline{K}_n,k)$ count, respectively, unlabeled list colorings and unlabeled proper colorings with $k$ colors.

Eventual equality conjecture. For each $n \in \mathbb{N}$, there is an $N \in \mathbb{N}$ such that

$$P_{\ell}(\overline{K}_n,k)=P(\overline{K}_n,k)$$

whenever $k \geq N$.

The conjecture concerns disconnected unlabeled graphs with repeated isomorphic connected components. The preceding discussion notes that equality can fail for small values of $k$, while asserting that the two counting functions should eventually agree for every fixed $n$.