Uniform lattice conjecture for the Lipschitz extension constant

Let $M$ be a metric space and let $\Gamma\subset M$ be a lattice. Say that $\Gamma$ has the $\mathcal{LE}$ if it is uniform: there exist an increasing function $\phi_{\Gamma}:\mathbb{R}_{+}\to\mathbb{R}_{+}$ and a constant $0<c\leq 1$ such that, for every $R>0$ and $m\in\Gamma$,

$$\nc\phi_{\Gamma}(R)\leq |\Gamma\cap B_R(m)|\leq\phi_{\Gamma}(R).$$

Uniform lattice conjecture. Every lattice $\Gamma\subset M$ with the $\mathcal{LE}$ is uniform. The source places this definition immediately before the conjectural discussion, but does not explicitly state a corresponding conjecture in the supplied span. This candidate therefore requires checking against the original surrounding text.