Bound conjecture for β-subgroups in finite simple groups

Let $r$ be a prime, let $L$ be a nonabelian simple group whose order is divisible by $r$, and let $x\in\operatorname{Aut}(L)$ be an automorphism of prime order. Bound conjecture for $\beta_r(x,L)$. For every such $L$ and $x$,

$$\beta_r(x,L)\leqslant\begin{cases} r, & \text{if } r\in\{2,3\},\\ r-1, & \text{if } r\geqslant 5. \end{cases}$$

The authors state that they know no counterexamples. The conjecture would improve the paper's general upper bound $2(r-2)$ in the case of odd primes and is the stronger statement from which the preceding conjecture follows.