The almost-regular orbit neighbourhood conjecture for generalised Saxl graphs

Let $G\leqslant\operatorname{Sym}(\Omega)$ be a finite primitive permutation group with base size $b(G)\geqslant 2$. For $\alpha\in\Omega$, call a $G_\alpha$-orbit on $\Omega$ almost-regular if it contains a point of some base of size $b(G)$ containing $\alpha$; let $N(\beta)$ denote the neighbourhood of $\beta$ in the generalised Saxl graph $\Sigma(G)$. Almost-regular orbit neighbourhood conjecture. For any $\alpha,\beta\in\Omega$, the neighbourhood $N(\beta)$ meets every almost-regular $G_\alpha$-orbit. This extends the Burness–Huang conjecture from base-two groups to arbitrary primitive groups with base size at least two; no resolution is supplied in the source.