On a relation between the Sylow and Baer–Suzuki theorems

Given a set $\pi$ of primes, say that a finite group $G$ satisfies the Sylow $\pi$-theorem if every two maximal $\pi$-subgroups of $G$ are conjugate; equivalently, the full analog of the Sylow theorem holds for $\pi$-subgroups. Say also that a finite group $G$ satisfies the Baer–Suzuki $\pi$-theorem if every conjugacy class of $G$ every pair of whose elements generate a $\pi$-subgroup itself generates a $\pi$-subgroup. In this article we prove, using the classification of finite simple groups, that if a finite group satisfies the Sylow $\pi$-theorem then it satisfies the Baer–Suzuki $\pi$-theorem as well.