Bonamy–Bousquet's conjecture on color savings for graph powers

Let $G$ be a graph with maximum degree $\Delta$, let $k\geqslant 2$, and let $G^k$ denote its $k$-th power. Write $f(k,\Delta)$ for the corresponding upper-bound function on the number of colors.

Bonamy–Bousquet's conjecture. For every $k\geqslant 2$, only finitely many graphs $G$ satisfy

$$\chi(G^k)\geqslant f(k,\Delta)+1-k.$$

This conjecture proposes that, apart from finitely many exceptional graphs for each $k$, one can spare $k$ colors from the naive upper bound for coloring graph powers. It strengthens the known result that for $k\geqslant 3$, $\chi(G^k)\leqslant f(k,\Delta)-1$.