The weak gap conjecture for HK, Følner, isoperimetric and spectral-density types

Let $G$ be a finitely generated amenable group, and let $P(n)$, $\mathcal F(n)$, $\Lambda(n)$ and $\mathcal N(\lambda)$ denote respectively the heat-kernel, Følner, isoperimetric-profile and spectral-density functions. Weak gap conjecture of type HK, F, IP or SD. There exists a positive parameter $\beta$ (respectively $\gamma$, $\rho$ or $\delta$) such that the corresponding gap conjecture of type HK (respectively F, IP or SD) with that parameter holds. The source introduces this as a weaker existence assertion for one of the four gap-conjecture types; it does not report a resolution.