On the sharp Baer–Suzuki theorem for the $\pi$-radical: sporadic groups

Let $\pi$ be a proper subset of the set of all primes and ${|\pi|\geq 2}$. Denote the smallest prime not in $\pi$ by $r$ and let $m=r$ if $r=2,3$, and $m=r-1$ if $r\geq 5$. We study the following conjecture: A conjugacy class $D$ of a finite group $G$ lies in the $\pi$-radical $\mathrm{O}_\pi(G)$ of $G$ if and only if every $m$ elements of $D$ generate a $\pi$-subgroup. We confirm this conjecture for the groups $G$ whose every nonabelian composition factor is isomorphic to a sporadic or alternating group.