The Gap Conjecture for finitely generated group growth

Let $G$ be a finitely generated group with growth function $\gamma_G(n)$. Write $f\prec g$ when $f$ is strictly bounded above by $g$ in the relevant growth comparison. Gap Conjecture. If

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

then $G$ has polynomial growth. This conjecture proposes a sharp boundary between polynomial and intermediate growth; the source presents it as an open problem, with $e^{\sqrt n}$ motivated by known lower bounds for groups of intermediate growth.