Gap conjectures for heat kernel, Følner function, isoperimetric profile and spectral density

Let $G$ be a finitely generated amenable group, with the heat-kernel function $P(n)$, Følner function $\mathcal F(n)$, isoperimetric profile $\Lambda(n)$, and spectral density $\mathcal N(\lambda)$ as defined in the source. Let $\beta,\gamma,\rho,\delta$ be positive parameters. Gap conjectures with parameters. The following alternatives should hold: $P(n)$ has power-rate decay $n^{-d/2}$ or satisfies $P(n)\preceq e^{-n^\beta}$; $\mathcal F(n)$ has polynomial growth $n^d$ or grows at least as fast as $e^{n^\gamma}$; $\Lambda(n)$ has power-rate decay $n^{-d/2}$ or satisfies $\Lambda(n)\succeq(\log n)^{-\rho}$; and $\mathcal N(\lambda)$ has power decay of type $\lambda^{d/2}$ near zero for some $d\in\mathbb N$, or satisfies $\mathcal N(\lambda)\succeq e^{-\lambda^{-\delta}}$ near zero. These conjectures concern gaps between polynomial behavior and slower-than-polynomial behavior; the polynomial cases are known for groups of polynomial growth, but the source does not resolve the proposed alternatives.