AAF17's first post-critical homotopy-type conjecture for spheres

For $n\in\mathbb{Z}_{>0}$, let $\mathbb{S}^n$ be the unit $n$-sphere, let $\mathrm{SO}(n+1)$ be the special orthogonal group, and let $A_{n+2}$ be the alternating group of degree $n+2$. Write $*$ for the topological join.

AAF17's sphere conjecture. There exists an $\varepsilon>0$ such that

$$\mathrm{VR}_{r}(\mathbb{S}^n)\simeq\mathbb{S}^n*\big(\mathrm{SO}(n+1)/A_{n+2}\big)$$

for every $r\in\left(\arccos\left(\frac{-1}{n+1}\right),\arccos\left(\frac{-1}{n+1}\right)+\varepsilon\right)$.

The conjecture is known for $n=1$ and $n=2$, while the source states that it remains open in general. It predicts the first homotopy type after the critical Vietoris-Rips scale for spheres.