Existence of a NIP ω-categorical theory with an invariant type having uncountably many Aut-conjugates

Determine whether there exists a NIP ω-categorical theory T with countable model M and a type p(x) over M that is invariant under Aut(M/acl^{eq}(∅)) but has uncountably many Aut(M)-conjugates. Establishing the existence or nonexistence of such a type would clarify whether the classification of invariant Keisler measures in Corollary 7 (which decomposes measures via Aut(M)-conjugacy classes of Aut(M/acl^{eq}(∅))-invariant types) holds in full generality for NIP ω-categorical theories.

Background

In Section 7 the authors show that, under the Invariant Extension Property (IEP) and assuming no Aut(M/acl^{eq}(∅))-invariant type has uncountably many Aut(M)-conjugates, every invariant Keisler measure on an ω-categorical structure decomposes as an integral over Aut(M)-conjugacy classes of such invariant types (Corollary \ref{cor:IEPchar}).

They prove that IEP holds for all NIP ω-categorical theories, so the remaining hypothesis concerns the size of Aut(M)-conjugacy classes of Aut(M/acl^{eq}(∅))-invariant types. The authors explicitly state that they do not know any NIP ω-categorical examples where such a type has uncountably many Aut(M)-conjugates, noting also the lack of known non-G-finite homogeneous structures in a finite language.

Resolving this existence question would determine whether their classification of invariant Keisler measures applies to all NIP ω-categorical theories without additional assumptions.