Non-perfectness conjecture for generating graphs of finite simple groups

Let $G$ be a finite non-abelian simple group. Its generating graph $\Gamma(G)$ has the elements of $G$ as vertices, with two vertices adjacent when they generate $G$. A graph is perfect if every induced subgraph has chromatic number equal to its clique number.

Non-perfectness conjecture. The generating graph $\Gamma(G)$ is not perfect.

The conjecture is motivated by the result that $\Gamma(A_n)$ and $\Gamma(S_n)$ are perfect exactly when $n<5$, while $A_5$ is the smallest 2-generated finite group whose generating graph is not perfect. The conjecture remains open in the source.