The generalized Sims conjecture for finite primitive groups

Let $G$ be a finite primitive group acting on a set $\a0\Omega$, let $\alpha\in\Omega$, and let $G_\alpha$ be the stabilizer of $\alpha$. For a point $\beta\in\Omega$, write $G_{\alpha\beta}=G_\alpha\cap G_\beta$ and let $\beta^{G_\alpha}$ be the suborbit of length $d>1$. If $\Gamma$ is the corresponding orbital graph, define

$$G_{\alpha}^{+[1]}=\bigcap_{\delta\in\Gamma^+(\alpha)}G_{\alpha\delta}.$$

Generalized Sims conjecture. There exists a function $g:\mathbb{N}\to\mathbb{N}$ such that, if $G$ is a finite primitive group with a suborbit $\beta^{G_\alpha}$ of length $d>1$, then either

$$|G_{\alpha\beta}|\leq g\left(|G_{\alpha\beta}:G_{\alpha}^{+[1]}|\right),$$

or $G$ belongs to a well-described and well-determined list of exceptions. This is proposed as a strengthening of Sims's conjecture, which bounds the order of a point stabilizer in terms of the suborbit length. A positive solution would refine the structural information about point stabilizers and arc stabilizers in finite primitive groups; the source does not give a complete resolution, although its status evidence indicates that the conjecture is resolved.