The two-element base conjecture for saturated fusion systems

Let $\mathcal{F}$ be a saturated fusion system on a finite $p$-group $P$. A subset $\Delta\subseteq P$ is a base of $\mathcal{F}$ if there exists a morphism $\varphi$ in $\mathcal{F}$ such that $\varphi(\langle\Delta\rangle)$ is fully $\mathcal{F}$-centralized and the centralizer fusion system $\mathcal{C}:=\mathrm{C}_{\mathcal{F}}(\varphi(\langle\Delta\rangle))$ is trivial, that is,

$$\mathcal{C}=\mathcal{F}_{\mathrm{C}_P(\Delta)}(\mathrm{C}_P(\Delta)).$$

The two-element base conjecture for saturated fusion systems. Every saturated fusion system has a base of size $2$.

This conjecture strengthens the finite-group $p$-base conjecture because block fusion systems can be viewed through this framework, while exotic fusion systems are not necessarily attached to finite groups. The paper presents it as an open strengthening.