Generalized Cranston–Kim conjecture for powers of graphs

Let $G$ be a simple connected graph with maximum degree $\Delta\geq 3$. For $k\in\mathbb{N}^*$, let $G^k$ be the graph obtained by joining vertices at distance at most $k$, and define

$$D(k,\Delta)=\Delta\sum_{i=1}^{k}(\Delta-1)^{i-1}=\Delta\frac{(\Delta-1)^k-1}{\Delta-2}.$$

A Moore graph is a graph on $\Delta^2+1$ vertices whose square is a clique. Generalized Cranston–Kim conjecture. For any $k\in\mathbb{N}^*$, except for Moore graphs when $k=2$, the $k$th power $G^k$ is $(D(k,\Delta)-1)$-choosable. The paper proves this assertion for $k\geq 3$, and the $k=2$ case was proved subsequently, so the conjecture is solved.