Diameter-preservation conjecture for the Vietoris–Rips thickening retraction

Let $k\geq 1$, let $r$ satisfy

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

let $\mu\in\mathrm{VR}^m(S^1;r)$, and let $\iota$ map the boundary of the Barvinok–Novik orbitope $\mathcal{B}_{2k}$ into the Vietoris–Rips metric thickening. Let $p$ be radial projection to $\partial\mathcal{B}_{2k}$ and let $\mathrm{SM}_{2k}$ be the spherical moment map. Diameter-preservation conjecture.

$$\operatorname{diam}(\operatorname{supp}(\mu))=\operatorname{diam}\bigl(\operatorname{supp}(\mu)\cup\operatorname{supp}(\iota\circ p\circ\mathrm{SM}_{2k}(\mu))\bigr).$$

This would control the effect of the proposed retraction on supports of measures and is intended to help show that $p\circ\mathrm{SM}_{2k}$ and $\iota$ are homotopy inverses. The supplied text does not state a resolution.