The two-element commutative p-base conjecture

Let $G$ be a finite group and let $p$ be a prime. If $P$ is a Sylow $p$-subgroup of $G$, a subset $\Delta\subseteq P$ is a $p$-base when $\mathrm{C}_G(\Delta)$ is $p$-nilpotent, meaning that it has a normal $p$-complement.

The two-element commutative $p$-base conjecture. Every finite group has a commutative $p$-base of size $2$ for every prime $p$.

The conjecture seeks a uniform bound on the number of elements needed to control centralizers in finite groups; the paper proves the corresponding bound for $p$-solvable groups and verifies the conjecture in several further classes, but does not establish it in general.