Grigorchuk's conjecture $G^*(\beta)$ for non-virtually-nilpotent groups

Let $G$ be a finitely generated group with growth function $\gamma_G(n)$, and let $\beta$ be the parameter used in the stronger form $G^*(\beta)$ of the parameterized Gap Conjecture. A group is virtually nilpotent if it has a nilpotent subgroup of finite index. Conjecture $G^*(\beta)$. If $G$ is not virtually nilpotent, then

$$\gamma_G(n)\succeq e^{n^\beta}.$$

This strengthens the corresponding gap assertion by giving a lower bound for every non-virtually-nilpotent group; the source does not state whether it is resolved.