The locally virtually free group conjecture for arbitrary $G$-TCF

Let $G$ be a group. The theory $G$-$\operatorname{TCF}$ is the model companion theory for fields equipped with an action of $G$. A group is locally virtually free if every finitely generated subgroup is virtually free. The locally virtually free group conjecture. The theory $G$-$\operatorname{TCF}$ exists if and only if $G$ is locally virtually free. This extends the finitely generated proposal to arbitrary groups, incorporating both virtually free groups and the additive group $\mathds{Q}$, for which $\mathds{Q}$-ACFA is known to exist. The general assertion remains open in the supplied text.