Yang–Revin–Vdovin conjecture on conjugates detecting the pi-radical

Let $\pi$ be a fixed subset of the set $\mathbb{P}$ of all primes with $\pi'=\mathbb{P}\setminus\pi\ne\varnothing$. A $\pi$-group is a finite group whose order has all prime divisors in $\pi$. Set

$$r=r(\pi)=\min\pi'$$

and

$$m=m(\pi)=\begin{cases}r,&r\in\{2,3\},\r-1,&r\geqslant 5.\end{cases}$$

For a finite group $G$, let $\operatorname{O}_\pi(G)$ denote its $\pi$-radical, the largest normal $\pi$-subgroup of $G$. Yang–Revin–Vdovin conjecture. In every finite group $G$, the $\pi$-radical $\operatorname{O}_\pi(G)$ coincides with the set

$$\{x\in G\mid \langle x^{g_1},\dots,x^{g_m}\rangle\text{ is a }\pi\text{-group for all }g_1,\dots,g_m\in G\}.$$

This conjecture proposes a sharp Baer–Suzuki-type characterization of the $\pi$-radical using finitely many conjugates of an element. The paper invokes it while studying the case of almost simple groups with socle $\,{}^2F_4(q^2)'$, but does not establish the conjecture in full generality.