Let $G$ be a nonabelian finite simple group, and let $p$ and $q$ be
distinct prime divisors of the order of $G$. Is it true that the
number of elements of $G$ of order $p$ never equals the number of elements
of $G$ of order $q$?

*Remark:* My husband ran a [GAP][1] computation checking all nonabelian finite simple groups
of order less than 30000000, and did not find a counterexample.

  [1]: https://www.gap-system.org/