Cameron–Sheehan–Spiga conjecture on semiregular automorphisms of cubic vertex-transitive graphs

Let $\Gamma$ be a finite, connected graph with no loops or multiple edges. It is cubic if it is regular of valency $3$, and vertex-transitive if $\operatorname{Aut}(\Gamma)$ acts transitively on its vertices. An automorphism is semiregular if, in its action on the vertices, all cycles in its disjoint cycle decomposition have the same length.

Cameron–Sheehan–Spiga conjecture. There exists a function $f:\mathbb{N}\to\mathbb{N}$ with

$$\lim_{n\to\infty} f(n)=\infty$$

such that, if $\Gamma$ is a cubic vertex-transitive graph with $n$ vertices, then $\Gamma$ has a semiregular automorphism of order at least $f(n)$.

The conjecture predicts that the order of a semiregular automorphism in a cubic vertex-transitive graph must become arbitrarily large as the number of vertices grows. The source states that it holds when $\Gamma$ is either a Cayley graph or an arc-transitive graph, while the general case is presented as open.