Hrushovski's nonexistence conjecture for groups containing a copy of $\mathbb{Z}\times\mathbb{Z}$

Let $G$ be a group, and let $G-\mathrm{TCF}$ denote the theory of existentially closed $G$-fields, when such a theory exists. Hrushovski's nonexistence conjecture. If $G$ has a subgroup isomorphic to

$$\mathbb{Z}\times\mathbb{Z},$$

then the theory $G-\mathrm{TCF}$ does not exist. This was proposed as an attempted generalization of the preceding nonexistence result for $C_p^{(\omega)}$; the supplied text gives no resolution of the claim.