Casselgren's random-list colouring conjecture for graphs of unbounded degree

Given positive integers $k \leq m$, a random $(k,m)$-list-assignment assigns independently to each vertex a uniformly random $k$-element subset of $\{1,\ldots,m\}$. An $n$-vertex graph is $(k,m)$-colourable when it has a proper colouring choosing the colour of every vertex from its assigned list. Let $k \geq 2$ be fixed, and let $G$ be an $n$-vertex graph with maximum degree $\Delta:= \Delta(n)$. Casselgren's conjecture. If $m \gg n^{1/k^2}\Delta^{1/k}$ and $m \gg \Delta$, then $G$ is asymptotically almost surely $(k,m)$-colourable. This extends the known threshold for bounded-degree graphs to graphs whose maximum degree may grow with $n$; the case $k=2$ is known, as is the bounded-degree case, while the general conjecture remains open.