Metric thickening sphere-spectrum conjecture for the circle

Let $S^1$ be the unit circle, let $r>0$, and let $\mathrm{VR}^m(S^1;r)$ denote its Vietoris–Rips metric thickening. For each integer $k\geq 1$, let $\mathcal{B}_{2k}$ be the Barvinok–Novik orbitope and let $S^{2k-1}$ be the $(2k-1)$-dimensional sphere. Metric thickening sphere-spectrum conjecture. For

$$\frac{2\pi(k-1)}{2k-1}\le r<\frac{2\pi k}{2k+1},$$

the metric thickening $\mathrm{VR}^m(S^1;r)$ is homotopy equivalent to the boundary of $\mathcal{B}_{2k}$, and hence to $S^{2k-1}$. The conjecture would identify the homotopy types of the Vietoris–Rips metric thickenings of the circle across the successive parameter ranges; the corresponding statement for Vietoris–Rips simplicial complexes is known, while the metric-thickening version is presented as conjectural here.