On the conjugacy class exponent of the finite simple groups

The generalized order $e_G(g)$ of an element $g$ of a group $G$ is the smallest positive integer $k$ such that there exist $x_1,\dots ,x_k\in G$ such that $g^{x_1}\dots g^{x_k}=1$, where $g^x=x^{-1}gx$. Let $e(G)=\max \{e_G(g)\mathrm{\mid }g\in G\}$. We provide upper bounds for $e(G)$ for every finite simple group $G$. In particular, we show that $e(G)\le 8$ unless $G\in \{PS{L}_n(q),PS{U}_n(q),E_6(q),{}^2{E}_6(q)\}$. For the latter groups $e(G)\le n,3n+3,36,36$, respectively. In addition, we bound from above the generalized order of semisimple and unipotent elements of finite simple groups of Lie type.