The list color function threshold conjecture for complete bipartite graphs

Let $K_{2,n}$ be the complete bipartite graph with parts of sizes $2$ and $n$, and let $\tau(G)$ denote the list color function threshold of a graph $G$.

List color threshold conjecture.

$$\tau(K_{2,n}) = \Theta(\sqrt{n}) \quad\text{as } n \to \infty.$$

A lower bound of order $\sqrt{n}$ is known, while this conjecture asserts a matching upper bound and therefore determines the asymptotic growth of the threshold.