The Weak Gap Conjecture for group growth

Let $G$ be a finitely generated group with growth function $\gamma_G(n)$, and let $\beta$ satisfy $0<\beta<1$. Weak Gap Conjecture. There exists such a $\beta$ for which

$$\gamma_G(n)\prec e^{n^\beta}$$

implies that the Gap Conjecture with parameter $\beta$ holds. This is a weaker formulation introduced alongside the parameterized Gap Conjectures; the source gives no resolution.