The orbit-equivalent primitive closed permutation groups conjecture

Let $G$ and $H$ be distinct orbit-equivalent primitive closed permutation groups on a countably infinite set $X$. The groups are orbit-equivalent when they have the same orbits on finite subsets of $X$. They should both belong to the list

$$\operatorname{Aut}(X,<),\quad \operatorname{Aut}(X,B),\quad \operatorname{Aut}(X,C),\quad \operatorname{Aut}(X,S),\quad \operatorname{Sym}(X).$$

Orbit-equivalent groups conjecture. $G$ and $H$ belong to the list above. This would strengthen the main theorem by removing its assumption that the groups are not 2-transitive. The result is open in the source.