Nonexistence of cross-characteristic composition factors in isospectral groups

Let $L\in\mathcal{L}$, where $\mathcal{L}$ is the specified list of nonabelian simple groups, let $S$ be a finite simple group of Lie type over a field whose characteristic differs from the defining characteristic of $L$, and let $G$ be a finite group. Write $\omega(G)$ for the set of element orders of $G$. Cross-characteristic composition-factor conjecture. If

$$\omega(G)=\omega(L),$$

then $S$ is not a composition factor of $G$. This conjecture is stated as the special case whose validity would imply almost recognizability for every group in $\mathcal{L}$; its resolution is not given in the supplied source.