Stronger finite-exception conjecture for choosability of graph powers

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

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

Stronger finite-exception conjecture. For any $k\in\mathbb{N}^*$, except for a finite number of graphs, the $k$th power $G^k$ is $(D(k,\Delta)+1-k)$-choosable. This is posed as a stronger generalization of the square-choosability conjecture, and the source does not report a resolution.