Finiteness of essential subgroups in G-finite oligomorphic groups

Determine whether every G-finite oligomorphic group has only finitely many conjugacy classes of essential subgroups, where an essential subgroup is an almost essential open subgroup of minimal depth as defined in Definition 5.11.

Background

The paper introduces essential subgroups: open subgroups that are almost essential (their conjugacy class generates the filter of open subgroups) and have minimal depth among such subgroups. It is shown that every G-finite oligomorphic group has at least one essential subgroup (Proposition 5.14).

The authors ask whether the number of conjugacy classes of essential subgroups must be finite in the G-finite setting. They note in Remark 5.16 that a more general question about almost essential subgroups of fixed depth has a negative answer (work in progress), but this does not settle the specific finiteness question for essential subgroups.