Smoothness of isomorphism for all oligomorphic groups

Determine whether the topological isomorphism relation on the class of all oligomorphic closed subgroups of Sym(ω) is smooth, i.e., Borel reducible to equality on the real numbers.

Background

Prior work showed that topological isomorphism on oligomorphic groups is Borel and lies below a countable Borel equivalence relation, but left open whether it is much lower—specifically, whether it is smooth (i.e., reducible to identity on ℝ).

This paper develops a general criterion (Theorem 4.7) that yields smoothness for two substantial Borel subclasses: oligomorphic groups without algebraicity, and those with finitely many essential subgroups up to conjugacy (Theorem 1.3). The full class, however, remains unresolved.