Conjecture on cross-characteristic composition factors in isospectral groups

Let $q$ be odd, and let $L$ be one of the following simple groups:

• $L_n(q)$, where $6\leqslant n\leqslant 26$ and $n$ is not prime;
• $U_n(q)$, where $5\leqslant n\leqslant 26$;
• $O_{2n}^+(q)$, where $5\leqslant n\leqslant 18$;
• $O_{2n}^-(q)$, where $5\leqslant n\leqslant 17$ and $n\neq 8,16$;
• $S_{2n}(q)$ and $O_{2n+1}(q)$, where $5\leqslant n\leqslant 15$ and $n\neq 8$.

If $G$ is a finite group isospectral to $L$, let $S$ be its unique nonabelian composition factor.

Cross-characteristic composition-factor conjecture. The group $S$ cannot be a group of Lie type in characteristic coprime to $q$.

The source records partial results excluding several families and notes that recognition is already solved for some cases; the full assertion remains unresolved in the supplied text.