Amenable strongly transitive action on a totally ordered set

Let $G$ be a group acting strongly transitively on a totally ordered set $X$ by order-preserving bijections. Amenability conjecture. There exists an amenable group $G$ admitting such an action on $X$. The conjecture concerns the existence of amenable groups with highly transitive order-preserving actions, in the context of higher Liouville properties. The source provides no resolution status or further evidence.