Burning Hamming graphs

The Hamming graph $H(n,q)$ is defined on the vertex set $[q]^n$ and two vertices are adjacent if and only if they differ in precisely one coordinate. Alon \cite{Alon} proved that the burning number of $H(n,2)$ is $\lceil\frac n2\rceil+1$. In this note we give a short proof of a fact that the burning number of $H(n,q)$ is $(1-\frac 1q)n+O(\sqrt{n\log n})$ for fixed $q\geq 2$ and $n\to\infty$.