The base-two conjecture for nilpotent subgroups of finite simple groups

Let $G$ be a finite non-abelian simple group and let $H$ be a non-trivial nilpotent subgroup. Regard $G$ as a permutation group on the cosets of $H$, and write $b(G,H)$ for the minimum number of conjugates of $H$ with trivial intersection.

Vdovin's base-two conjecture. One should have

$$b(G,H)=2.$$

Zenkov's theorem gives the general upper bound $b(G,H)\leqslant 3$ for non-trivial nilpotent subgroups in finite almost simple groups. The asserted improvement to $2$ in the simple-group case was stated as an open problem and was later proved by Mazurov and Zenkov in 1996.