On the intersections of solvable Hall subgroups in finite groups

The following conjecture is considered: if a finite group $G$ possesses a solvable $\pi$-Hall subgroup $H$, then there exist elements $x,y,z,t\in G$ such that the identity $H\cap H^x\cap H^y\cap H^z\cap H^t=O_\pi(G)$ holds. Under additional conditions on $G$ and $H$, it is shown that a minimal counterexample to this conjecture must be an almost simple group of Lie type.