A Huppert-type $ρ^*$--$σ^*$ theorem for sub-class sizes

For a finite group $G$, let $ρ^{*}(G)$ be the set of primes dividing some sub-class size of $G$, and let $σ^{*}(G)$ be the maximum number of distinct prime divisors of a sub-class size. Motivated by the conjugacy-class version of Huppert's $ρ$--$σ$ problem, we prove that \[ |ρ^{*}(G)|<2σ^{*}(G) \]