Near-$r_n$ homotopy-type conjecture for Vietoris–Rips complexes of spheres

Let $S^n$ be the $n$-sphere, let $r_n$ be the scale defined in the paper, let $\mathrm{SO}(n+1)$ be the special orthogonal group, let $A_{n+2}$ be the alternating group, and let $*$ denote the join. Near-$r_n$ conjecture. For every $n$, there exists $\varepsilon>0$ such that

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

for all $r_n<r<r_n+\varepsilon$. This is a local prediction for the homotopy type immediately above the scale $r_n$; the conjecture remains open.