Homological detection conjecture for Vietoris–Rips complexes of integer lattices

Let $\mathbb{Z}^n$ have the Manhattan metric, let $\{0,1\}^n\subseteq\mathbb{Z}^n$ have the induced metric, and let $\widetilde{H}_i(-;\mathbb{Z})$ denote reduced homology with integer coefficients. Homological detection conjecture. For all relevant $i$, $n$, and $r$,

$$\widetilde{H}_i\bigl(\mathcal{VR}(\mathbb{Z}^n;r);\mathbb{Z}\bigr)\neq 0$$

if and only if

$$\widetilde{H}_i\bigl(\mathcal{VR}(\{0,1\}^n;r);\mathbb{Z}\bigr)\neq 0.$$

The inclusion of the discrete cube into the integer lattice is known to induce an injective map on reduced homology, so the conjecture asks whether it also detects all nonvanishing reduced homology of the lattice Vietoris–Rips complex.