2 problems

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Mazurov's recognizability conjecture for simple groups with disconnected prime graph
Mazurov's conjecture. If $G$ is not isomorphic to $A_6$ and $\Gamma(G)$ has at least three connected components, then $G$ is recognizable by its spectrum.
Conjecturesolvedfinite-groups prime-graphs recognitionarXiv posted 2 days ago
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The two-orders characterization conjecture for finite simple groups
Two-orders characterization conjecture. $G\cong S$ if and only if
Conjectureopenfinite-groups finite-simple-groups recognitionarXiv posted 2 days ago

Mazurov's recognizability conjecture for simple groups with disconnected prime graph