Thompson's conjecture for simple groups with connected prime graph

We deal with finite simple groups $G$ with the property $\pi(G)\subseteq\{2,3,5,7,11,13,17\}$, where $\pi(G)$ is the set of all prime divisors of the order of the group $G$. The set of all such groups is denoted by $\zeta_{17}$. A conjecture of Thompson in [Unsolved Problems in Group Theory, The Kourovka Notebook, 17th edn., Institute of Mathematics SO RAN, Novosibirsk (2010), Question 12.38] is proved valid for all groups with connected prime graph in $\zeta_{17}$.