For a finite group $G$ one defines the *spectrum* $\omega(G)$ as the set of element orders of $G$. The set $\omega(G)$ is uniquely determined by the subset $\mu(G)$ consisting of those elements of $\omega(G)$ that are maximal with respect to the divisibility relation.

I would like to know whether for a finite non-abelian simple group $G$ one has $|\mu(G)| \geq 3$.

If this is the case, is there a proof that works for all finite simple groups of Lie type (without treating each family differently)?