Lisi–Sabatini conjecture for simple groups

Let $G$ be a finite simple group, let $\pi(G)$ be the set of prime divisors of $|G|$, and for each $p \in \pi(G)$ let $H_p$ be a Sylow $p$-subgroup of $G$.

Lisi–Sabatini conjecture for simple groups. There exists an element $x \in G$ such that

$$H_p \cap H_p^x = 1$$

for all $p \in \pi(G)$.

This is the simple-group consequence of the Lisi–Sabatini conjecture together with the cited theorem of Mazurov and Zenkov. The paper's abstract indicates that the Lisi–Sabatini conjecture is proved for all non-alternating simple groups, and earlier work handles the relevant alternating groups, so this consequence is established in the paper's setting.