The Burness–Giudici conjecture on common neighbours in Saxl graphs

Let $G$ be a finite primitive permutation group acting on a finite set $\Omega$. A base is a subset of $\Omega$ whose pointwise stabiliser is trivial, and let $b(G)$ be the minimum size of a base. If $b(G)=2$, define the Saxl graph $\Sigma(G)$ to have vertex set $\Omega$, with two vertices adjacent precisely when they form a base for $G$.

Burness–Giudici conjecture. Every two vertices of $\Sigma(G)$ have a common neighbour.

This conjecture concerns the connectivity properties of Saxl graphs of primitive permutation groups with base size two. The paper studies the conjecture for primitive groups with socle a simple rank-one group of Lie type, including groups with socle $\operatorname{Ree}(q)$ and $\operatorname{Sz}(q)$; its general status is not resolved in the supplied text.