Do small or slender automorphisms force mixed identities?

Determine whether, for an ω‑categorical relational structure (X,R), the existence of a non‑trivial small automorphism (i.e., an automorphism fixing an infinite definable subset pointwise) or a non‑trivial slender automorphism (i.e., an automorphism c for which there exist a finite set Y and an infinite definable set Z ⊆ X with Z ⊆ {x ∈ X | x.c ∈ acl(Y ∪ {x})}) implies that the automorphism group Aut(X,R) satisfies a mixed identity.

Background

The main theorem (Theorem 2.2.1) shows that any mixed identity in an ω-categorical structure must have a slender critical constant and, in algebraically convex settings, a small critical constant. This suggests a potential converse direction: does the presence of such automorphisms guarantee the existence of mixed identities?

The authors explicitly state that they do not know whether this converse holds, leaving a gap between necessary conditions (presence of slender/small elements) and sufficient conditions for mixed identities.