The game colouring number of powers of forests

We prove that the game colouring number of the $m$-th power of a forest of maximum degree $\mathrm{\Delta }\ge 3$ is bounded from above by $$\frac{(\mathrm{\Delta }-1{)}^m-1}{\mathrm{\Delta }-2}+2^m+1,$$ which improves the best known bound by an asymptotic factor of 2.