The extremely amenable subgroup conjecture for closed oligomorphic permutation groups

Let $G$ be a closed oligomorphic permutation group, meaning a closed permutation group with finitely many orbits on $n$-tuples for every $n\geq 1$. An expansion of an $\omega$-categorical structure is ordered when it includes a linear order, and is Ramsey when its age has the Ramsey property. The extremely amenable subgroup conjecture. Every closed oligomorphic permutation group contains a closed oligomorphic subgroup that is extremely amenable. Equivalently, every $\omega$-categorical structure has an ordered $\omega$-categorical expansion that is Ramsey. This is presented as an important variant of the unresolved conjecture discussed immediately beforehand, reformulated in terms of topological groups; no resolution is supplied in the source.