Burness–Giudici Common Neighbour Conjecture for Saxl graphs

Let $G\leq \operatorname{Sym}(\Omega)$ be a primitive permutation group with base size $b(G)=2$, and let $\Sigma(G)$ be its Saxl graph, whose vertices are the points of $\Omega$ and whose edges are the 2-element bases of $G$. Common Neighbour Conjecture. Any two vertices of $\Sigma(G)$ share a common neighbour. Consequently, $\operatorname{diam}(\Sigma(G))\leq 2$. This conjecture concerns the connectedness and diameter of Saxl graphs associated with primitive groups of base size two; its resolution is not established in the supplied source.