Generalised Common Neighbour Conjecture for Saxl hypergraphs

Let $G\leq \operatorname{Sym}(\Omega)$ be a finite permutation group with base size $b(G)\geq 2$, and let $\mathcal{H}(G)$ be its Saxl hypergraph; in the original graph formulation, use $\Sigma(G)$. Generalised Common Neighbour Conjecture. For any distinct points $\alpha,\beta\in\Omega$, there exist a third point $\gamma\in\Omega\setminus\{\alpha,\beta\}$ and edges $E_\alpha,E_\beta$ of $\mathcal{H}(G)$ such that $\alpha,\gamma\in E_\alpha$ and $\beta,\gamma\in E_\beta$. This formulation generalises the common-neighbour property from Saxl graphs to Saxl hypergraphs; the supplied source gives no resolution.