Vdovin's conjecture on intersections of nilpotent subgroups

Let $G$ be a finite simple group and let $H$ be a nilpotent subgroup of $G$.

Vdovin's conjecture. Then $H \cap H^x = 1$ for some $x \in G$.

This conjecture concerns whether every nilpotent subgroup of a finite simple group has a conjugate with trivial intersection. It has been resolved for alternating and sporadic groups; the paper's abstract states that its proof is completed using results for the remaining simple groups.