The positive-power lower-bound conjecture for intermediate growth

Let $G$ be a group of intermediate growth, and let $\gamma_G(n)$ be its growth function. Write $f(n)\succcurlyeq g(n)$ when $f$ is bounded below by $g$ up to the comparison convention used in the source.

Positive-power lower-bound conjecture. Then

$$\gamma_G(n)\succcurlyeq\exp(n^\alpha)$$

for some $\alpha>0$.

Such a lower bound has been established for large classes of groups, but not in general; obtaining it for every group of intermediate growth remains open.