On the intersections of nilpotent subgroups in finite groups with socle $L_3(q)$ or $U_3(q)$

Earlier, the author described up to conjugation all pairs $(A,B)$ of nilpotent subgroups $A$ and $B$ in a finite group $G$ with socle $L_2(q)$ for which $A\cap B^g\ne 1$ for any element $g$ of $G$. A similar description was obtained later by the author for primary subgroups $A$ and $B$ of a finite group $G$ with socle $L_n(2^m)$. In this paper, we describe up to conjugation all pairs $(A,B)$ of nilpotent subgroups $A$ and $B$ of a finite group $G$ with socle $L_3(q)$ or $U_3(q)$ for which $A\cap B^g\ne 1$ for any element $g$ of $G$. The obtained results confirm in the considered cases the hypothesis that for a finite simple non-Abelian group $G$ and its nilpotent subgroup $N$ there is an element $g\in G$ such that $N\cap N^g=1$ (Problem 15.40 from “The Kourovka Notebook”).