Grigorchuk–Pansu conjecture in terms of Dirichlet–Poincaré profiles

Let $G$ be a finitely generated group, and let $D\Lambda^p_G(n)$ denote its Dirichlet–Poincaré profile for $p\in[1,\infty)$. Grigorchuk–Pansu conjecture for Dirichlet–Poincaré profiles. Either there is some $d$ such that

$$D\Lambda^p_G(n)\simeq n^{1-\frac{1}{d}}$$

for all $p\in[1,\infty)$, or

$$D\Lambda^p_G(n)\gtrsim \frac{n}{\log(n)}$$

for all $p\in[1,\infty)$. This is presented as an equivalent reformulation of the conjectured polynomial-versus-exponential Følner-function dichotomy; the source gives no resolution status.