Involutions in finite simple groups as products of conjugates

Let $G$ be a finite non-abelian simple group, $C$ a non-identity conjugacy class of $G$, and ${\mathrm{\Gamma }}_C$ the Cayley graph of $G$ based on $C\cup C^{-1}$. Our main result shows that in any such graph, there is an involution at bounded distance from the identity.