Homology-dimension conjecture for Vietoris–Rips complexes of hypercube graphs

Let $\mathbb{I}_n$ denote the $n$-dimensional hypercube graph, and let $\mathcal{VR}(\mathbb{I}_n;r)$ be its Vietoris–Rips complex at scale $r$. For $r\in\{2,3\}$, write $\widetilde{H}_i(-;\mathbb{Z})$ for reduced homology in dimension $i$. Homology-dimension conjecture. For $n\geq r+2$,

$$\widetilde{H}_i(\mathcal{VR}(\mathbb{I}_n;r);\mathbb{Z})\neq 0$$

if and only if $i\in\{r+1,2^r-1\}$. This extends the established cases at scales $2$ and $3$, and predicts that these are the only dimensions supporting nontrivial homology for the indicated parameters.