The regular-orbit conjecture for cubic vertex-transitive graphs

Let $\Gamma$ be a cubic vertex-transitive graph of order $n$ that is not isomorphic to $K_{3,3}$ or a split Praeger–Xu graph, and let $g\in\operatorname{Aut}(\Gamma)$. A regular orbit of $g$ is an orbit whose size equals the order of $g$. The regular-orbit conjecture. At least

$$\frac{2}{3}n$$

vertices of $\Gamma$ lie on a regular orbit of $g$. The claim is motivated by a family of cubic vertex-transitive graphs in which two thirds of the vertices lie on regular orbits, and by the assertion that split Praeger–Xu graphs are the exceptional extreme case. Its resolution is not given in the source.