Lisi–Sabatini conjecture on simultaneous minimal intersections of Sylow subgroups

Let $G$ be a finite group with distinct prime divisors

$$\pi(G)=\{p_1,\ldots,p_k\},$$

and let $P_i$ be a Sylow $p_i$-subgroup of $G$.

Lisi–Sabatini conjecture. There exists an element $x \in G$ such that, for each $i$, $P_i \cap P_i^x$ is inclusion-minimal in the set

$$\{P_i \cap P_i^g: g \in G\}.$$

The conjecture generalizes the problem of finding conjugates with small intersections of Sylow subgroups. It is proved for metanilpotent groups of odd order and all sufficiently large alternating and symmetric groups, while the general case remains open.