The small sphere hypothesis implies the small gap property

Let $G$ be a finitely generated, virtually nilpotent group with the Small Sphere (SS) Property. For $r\in(0,1)$, define the small gap property (SG property) by requiring that there is some $\delta>0$ such that

$$\lim_{n\to\infty}\eta_n(G,\delta)>r,$$

where

$$\eta_n(G,\delta)=\min\{\rho(A,B):g_B(A)<\delta,\ A,B\text{ subballs of }G,\ |A|>n\}$$

and

$$\rho(A,B)=\left\{\frac{|C|}{|A|}:C\subseteq A\cap B\text{ is a subball}\right\}.$$

Small sphere to small gap conjecture. Then $G$ has the SG property. The SG property would provide the ball-intersection control needed for the SBM theory; it is known for $\mathbb{Z}^d$ with standard generators, but the source reports no verified non-abelian examples and leaves the general implication open.