Are all finitely generated groups defined by their types?

Determine whether every finitely generated group G is defined by its types; equivalently, prove or refute that for every finitely generated group G and every group H, if H and G are isotypic (i.e., they realize the same first-order types of tuples of elements), then H is isomorphic to G.

Background

The paper studies isotypical equivalence in groups, where two groups are isotypic if they realize the same first-order types of tuples of elements. A group G is said to be defined by its types if every group isotypic to G is isomorphic to G. This is a strong form of rigidity linking model-theoretic properties (types) to algebraic structure (isomorphism).

Prior work has established that several significant classes of finitely generated groups—including virtually polycyclic groups, finitely generated metabelian groups, and finitely generated rigid groups—are strongly defined by their types, meaning any elementary embedding between isotypic copies is an isomorphism. Despite these advances, it remains unresolved whether this phenomenon holds universally across all finitely generated groups.