Exceptional-groups conjecture for intersecting Saxl-hypergraph edges

Let $G\leq \operatorname{Sym}(\Omega)$ be a finite primitive permutation group with base size $b(G)\geq 2$, and let $\mathcal{H}(G)$ be its Saxl hypergraph. Exceptional-groups conjecture. Exactly one of the following holds: $G$ is $\operatorname{S}_n$ in its natural action with $n\geq 4$; $G$ is $\operatorname{A}_n$ in its natural action with $n\geq 6$; or, for every pair of distinct points $\alpha,\beta\in\Omega$, there exist edges $E_\alpha,E_\beta$ of $\mathcal{H}(G)$ such that $\alpha\in E_\alpha$, $\beta\in E_\beta$, and $|E_\alpha\cap E_\beta|=1$. The claim asserts that the symmetric and alternating natural actions are the only exceptions to this edge-intersection property; the supplied source gives no resolution.