Mazurov's conjecture on finite groups isospectral to simple groups

Let $L$ be a finite nonabelian simple group, and let $G$ be a finite group. The groups $G$ and $L$ are isospectral when $\omega(G)=\omega(L)$, where $\omega(H)$ denotes the set of element orders of a finite group $H$. Mazurov's conjecture. For every finite nonabelian simple group $L$, apart from a finite number of sporadic, alternating and exceptional groups and apart from several series of classical groups of small dimensions, if a finite group $G$ is isospectral to $L$ then $G$ is an almost simple group with socle isomorphic to $L$. This conjecture describes the expected generic recognition of finite simple groups by their spectra; the stated exceptions include finitely many groups and several low-dimensional classical series, and the source does not specify a resolution of the conjecture.