The virtually free group conjecture for $G$-TCF

Let $G$ be a finitely generated group. The theory $G$-$\operatorname{TCF}$ is the model companion theory for fields equipped with an action of $G$. A group is virtually free if it has a free subgroup of finite index. The virtually free group conjecture. The theory $G$-$\operatorname{TCF}$ exists if and only if $G$ is virtually free. This conjecture proposes a model-theoretic characterization of finitely generated virtually free groups. The paper confirms the equivalence for finitely generated commutative groups; the general finitely generated case remains open in the supplied text.