Synchronization conjecture under the $(*)$ condition

For a prime $p$, let $O_p(G)$ be the intersection of all Sylow $p$-subgroups of $G$. Say that $G$ has $(*)_p$ if there exist two Sylow $p$-subgroups whose intersection is contained in $O_p(G)$, and say that $G$ has $(*)$ if it has $(*)_p$ for all primes $p$. Let $G$ be a finite group with $(*)$, and let $(P_i)_{i=1}^n$ be Sylow subgroups for distinct primes $p_1,\ldots,p_n$. Synchronization conjecture under $(*)$. There exists $x \in G$ such that

$$P_i \cap P_i^x = O_{p_i}(G), \qquad i=1,\ldots,n.$$

This is a stronger form of the synchronization conjecture in the special case where the relevant pairwise minimal intersections are the corresponding normal $p$-cores. The paper notes that the condition $(*)$ holds for every finite simple group, but does not resolve the conjecture in general.