Thompson's conjecture for some finite simple groups with connected prime graph

Let $n$ be an even number and either $q=8$ or $q>9$. We prove a conjecture of Thompson (Problem 12.38 in the Kourovka Notebook) for an infinite class of finite simple groups of Lie type. More precisely, if $S\in\{C_n(q),B_n(q)\}$, then every finite group $G$ for which $Z(G)=1$ and $N(G)=N(S)$ will be isomorphic to $S$. Note that $N(G)=\{n\colon G$ has a conjugacy class of size $n\}$. The main consequence of this result is showing the validity of $AAM$'s conjecture (Problem 16.1 in the Kourovka Notebook) for the groups under study.