Collapsibility-number 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 $r$. Let the collapsibility number of a simplicial complex mean the parameter used in the paper to measure its collapsibility. Collapsibility-number conjecture. For $n\geq r+1$, the collapsibility number of $\mathcal{VR}(\mathbb{I}_n;r)$ is

$$2^r.$$

The value is established in the paper for $r\in\{2,3\}$, and the boundary cases $r=1$ and $r=n-1$ are described explicitly; the conjecture proposes the same formula throughout the stated range.