On the diameter of Engel graphs

Given a finite group $G$, the Engel graph of $G$ is a directed graph $\mathrm{\Gamma }(G)$ encoding pairs of elements satisfying some Engel word. Namely, $\mathrm{\Gamma }(G)$ is the directed graph, where the vertices are the non-hypercentral elements of $G$ and where there is an arc from $x$ to $y$ if and only if $[x{,}_{n}y]=1$ for some $n\in \mathbb{N}$. From previous work, it is known that, except for a few exceptions, $\mathrm{\Gamma }(G)$ is strongly connected. In this paper, we give an absolute upper bound on the diameter of $\mathrm{\Gamma }(G)$, when $\mathrm{\Gamma }(G)$ is strongly connected.