The Gap Conjecture with parameter beta

Let $G$ be a finitely generated group with growth function $\gamma_G(n)$, and let $\beta$ satisfy $0<\beta<1$. Gap Conjecture with parameter $\beta$. If

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

then $G$ has polynomial growth. The case $\beta=1/2$ is the original Gap Conjecture; the source explains that parameters below $1/2$ give weaker claims and parameters above $1/2$ give stronger ones. Its resolution is not supplied in the source.