Non-collapsibility conjecture for random Vietoris–Rips complexes

Let $K\subseteq {\mathbb R}^d$ be a convex body with smooth boundary, and let the distribution be uniform on $K$. Write $r_n$ for the radius parameter and ${{\mathcal R}}(n,r_n)$ for the random Vietoris–Rips complex. Non-collapsibility conjecture. There exists a sequence $\rho_n\gg\left(\frac{\ln n}{n}\right)^{1/d}$ such that ${{\mathcal R}}(n,r_n)$ is asymptotically almost surely not collapsible whenever $r_n\leq\rho_n$.

This conjecture strengthens the preceding discussion from contractibility to collapsibility: although cop-win graphs yield collapsible Vietoris–Rips complexes, the random geometric graph is asymptotically almost surely not cop-win at suitable radii above the connectivity-scale order. The conjectured non-collapsibility remains open.