Almost recognition by spectrum for specified simple classical groups

For a finite group $G$, write $\omega(G)$ for its spectrum, the set of orders of its elements. Two finite groups are isospectral if their spectra coincide. A finite group is almost simple if it contains a nonabelian simple group between its socle and its automorphism group.

Vasil'ev–Grechkoseeva conjecture. Suppose that $L$ is one of the following nonabelian simple groups:

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

Then every finite group isospectral to $L$ is an almost simple group with socle isomorphic to $L$.

The claim is presented as a conjectural sharpening after the general Mazurov conjecture was proved. The supplied source does not establish a resolution of this specific statement.