Baer–Suzuki conjecture for the π-radical

Let $\pi$ be a proper subset of the set of all primes containing at least two elements, and let $r$ be the minimal prime not in $\pi$. The invariant $\operatorname{BS}(\pi)$ is defined in the paper. Baer–Suzuki conjecture for the $\pi$-radical.

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

The authors state that they know no counterexamples; this conjecture would sharpen the bound for $\operatorname{BS}(\pi)$ obtained in the paper, with the first assertion a corollary of the corresponding bound for $\beta_r(x,L)$.