The Saxl graph diameter-two conjecture for primitive base-two groups

Let $G$ be a finite primitive permutation group with base size $b(G)=2$, and let $\alpha0\Sigma(G)$ be its Saxl graph, whose vertices are the points of the permutation domain and whose edges join pairs forming a base. The Saxl graph diameter-two conjecture. Either $G$ is a Frobenius group and $\alpha0\Sigma(G)$ is complete, or

$$\operatorname{diam}(\Sigma(G))=2.$$

This conjecture asserts that, apart from the complete Saxl graphs arising from primitive Frobenius groups, primitive base-two groups have highly connected Saxl graphs. It is presented as an open problem in the source.