Induced 5-hole 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 5-hole is an induced cycle on five vertices.

Induced 5-hole conjecture. There exists a subset $X$ of $G$ such that the subgraph of $\Gamma(G)$ induced by $X$ is a 5-hole.

This is presented as a stronger version of the conjecture that the generating graph of every finite non-abelian simple group is not perfect. The source establishes the corresponding 5-hole for $A_n$ and $S_n$ when $n\geq 5$, but leaves the general case open.