Sylow subgroups for distinct primes and intersection of nilpotent subgroups

Let $G$ be a finite group and let $(P_i{)}_{i=1}^n$ be Sylow subgroups for distinct primes $p_1,\dots ,p_n$. We conjecture that there exists $x\in G$ such that $P_i\cap P_i^x$ is inclusion-minimal in $\{P_i\cap P_i^g:g\in G\}$ for all $i$. As a first step in this direction, we show that a finite group cannot be covered by (proper) Sylow normalizers for distinct primes. Then we settle the conjecture in two opposite situations: symmetric and alternating groups of large degree and metanilpotent groups of odd order. Applications concerning the intersections of nilpotent subgroups are discussed.