Critical-scale conjecture for Vietoris–Rips complexes of spheres

Let $S^n$ be the $n$-sphere, and write $\mathrm{VR}(S^n;r)$ for its Vietoris–Rips complex at scale $r\ge 0$. Critical-scale conjecture. There are countably many critical scales

$$0=r_0<r_1<r_2<r_3<\ldots<\pi$$

such that, for every $i$ and all $r_i<r<r'<r_{i+1}$, the inclusion

$$\mathrm{VR}(S^n;r)\hookrightarrow\mathrm{VR}(S^n;r')$$

is a homotopy equivalence. This would give a countable decomposition of the interval below $\pi$ into ranges on which the homotopy type is constant; the conjecture remains open.