Metric Vietoris–Rips thickening conjecture for spheres

Let $S^n$ be the $n$-sphere, let $r_n$ be the scale under consideration, and let $\mathrm{VR}^m_\leq$, $\mathrm{VR}^m_<$, $\mathrm{VR}_<$, and $\mathrm{VR}_\leq$ denote the corresponding metric and simplicial Vietoris–Rips constructions. Then $\mathrm{SO}(n+1)/A_{n+2}$ is the quotient of $\mathrm{SO}(n+1)$ by the alternating group $A_{n+2}$, and $\Sigma^{n+1}$ denotes the $(n+1)$-fold suspension. Sphere reconstruction conjecture. For all $n$, there exists an $\varepsilon>0$ such that

$$\mathrm{VR}^m_\leq(S^n;r)\simeq\Sigma^{n+1}\ \frac{\mathrm{SO}(n+1)}{A_{n+2}}$$

for all $r_n\le r<r+\varepsilon$, while

$$\mathrm{VR}^m_<(S^n;r)\simeq\mathrm{VR}_<(S^n;r)\simeq\mathrm{VR}_\leq(S^n;r)\simeq\Sigma^{n+1}\ \frac{\mathrm{SO}(n+1)}{A_{n+2}}$$

for all $r_n<r<r+\varepsilon$. The result would extend the computed homotopy type at the critical scale $r_n$ to a neighborhood of that scale and relate the metric and simplicial Vietoris–Rips constructions.