Casselgren–Häggkvist conjecture on random list coloring of balanced complete-bipartite line graphs

Let $D$ be an even natural number and let $G= obreak\frac{}{}\mathcal{L}(K_{\frac{D}{2}+1,\frac{D}{2}+1})$ be the line graph of a complete bipartite graph. Thus $G$ has $n=(\frac{D}{2}+1)^2$ vertices and maximum degree $D$. For each vertex $v\in V(G)$, choose $L(v)\in\binom{[\frac{D}{2}+1]}{k}$ uniformly and independently. Casselgren–Häggkvist conjecture. There is a constant $C$ such that, whenever $k\geq C\log n$,

$$\mathbbm{P}(G\text{ is }L\text{-colorable})=1-o_{n\to\infty}(1).$$

This conjecture was posed after Johansson asked for a threshold for random list coloring. It has since been proved by several papers, so the conjecture is solved.