Intersections of three nilpotent subgroups in a finite group

We complete the proof of the theorem that any nilpotent subgroups $A$, $B$, and $C$ of a finite group $G$ satisfy the inclusion $A\cap B^x\cap C^y\le F(G)$, where $F(G)$ is the Fitting subgroup of $G$ and $x$ and $y$ are some elements in $G$. When $A=B=C$, we get an affirmative answer to Questions 17.40 and 19.37 from The Kourovka Notebook. The proof uses the classification of finite simple groups.