Conjecture on recognition by spectrum for finite classical groups

Let $L$ be one of the following nonabelian simple groups: $L_n(q)$ with $n\geqslant5$; $U_n(q)$ with $n\geqslant5$ and $(n,q)\neq(5,2)$; $S_{2n}(q)$ with $n\geqslant3$, $n\neq4$ and $(n,q)\neq(3,2)$; $O_{2n+1}(q)$ with $q$ odd, $n\geqslant3$, $n\neq4$ and $(n,q)\neq(3,3)$; or $O_{2n}^\varepsilon(q)$ with $n\geqslant4$ and $(n,q,\varepsilon)\neq(4,2,+),(4,3,+)$. Here $\omega(G)$ denotes the set of element orders of a finite group $G$, and $L$ is almost recognizable by spectrum when every finite group isospectral to $L$ is almost simple with socle isomorphic to $L$.

Recognition conjecture for classical groups. Every finite group isospectral to $L$ is an almost simple group with socle isomorphic to $L$.

This is a strengthened, explicit form of the earlier asymptotic almost-recognizability claim, with the listed low-rank and small-parameter exceptions. The supplied text does not state whether this precise formulation has been resolved.