Intersections of three nilpotent subgroups of finite groups

Under study is the conjecture that for every three nilpotent subgroups $A$, $B$, and $C$ of a finite group $G$ there are elements $x$ and $y$ such that $A\cap B^x\cap C^y\le F(G)$, where $F(G)$ is the Fitting subgroup of $G$. We prove that a counterexample of minimal order to this conjecture is an almost simple group. The proof uses the classification of finite simple groups.