Recognition-by-spectrum conjecture for finite simple classical groups

Let $L$ be one of the following finite simple classical groups:

$$\begin{gathered} L_n(q),\quad n\geqslant5;\\ U_n(q),\quad n\geqslant5,\quad (n,q)\neq(5,2);\\ S_{2n}(q),\quad n\geqslant3,\quad n\neq4,\quad (n,q)\neq(3,2);\\ O_{2n+1}(q),\quad q\text{ odd},\quad n\geqslant3,\quad n\neq4,\quad (n,q)\neq(3,3);\\ O_{2n}^{\varepsilon}(q),\quad n\geqslant4,\quad (n,q,\varepsilon)\neq(4,2,+),(4,3,+). \end{gathered}$$

Here $q$ is a prime power, and $\operatorname{Aut}L$ denotes the automorphism group of $L$. Two finite groups are isospectral when they have the same set of element orders, denoted by $\omega(G)$.

Recognition-by-spectrum conjecture. Every finite group isospectral to $L$ is isomorphic to some group $G$ satisfying

$$L\leqslant G\leqslant \operatorname{Aut}L.$$

This asserts that each listed simple classical group is almost recognizable by spectrum. The surrounding discussion presents this as a proposed summary of the paper's recognition results; the claim is not established in the supplied text.