Pansu–Grigorchuk gap conjectures with parameter beta

Let $G$, $\mu$, $P(n)$, $F(n)$ and $\mathcal N(\lambda)$ be as in the corresponding gap conjectures. Pansu–Grigorchuk gap conjectures with parameter $\beta$. There exists a number $\beta>0$ such that, for each of the following three assertions, the indicated alternative holds: (i) $P(n)$ either has power-rate decay or satisfies

$$P(n)\preceq e^{-n^\beta};$$

(ii) $F(n)$ either has polynomial growth or grows at least as fast as $e^{n^\beta}$; (iii) $\mathcal N(\lambda)$ either has power decay of type $\lambda^{d/2}$ for some $d\in\mathbb N$ or satisfies

$$\mathcal N(\lambda)\preceq e^{-\lambda^{-\beta}}.$$

The paper presents this as a parameterized modification of the three preceding conjectures and leaves it open.