Cranston–Rabern list Borodin–Kostochka conjecture

Let $G$ be a graph, and let $\Delta(G)$ denote its maximum degree. A clique of size $\Delta(G)$ is a complete subgraph on $\Delta(G)$ vertices. Cranston–Rabern's list-coloring conjecture. Every graph $G$ with $\Delta(G)\geq 9$ and no clique of size $\Delta(G)$ is $(\Delta(G)-1)$-choosable. The supplied text says the conjecture is known for claw-free graphs and remains open when $9\leq\Delta(G)\leq 68$ under the same condition.