Spherical distance-map conjecture for spheres

Let $n\geq 1$ be an integer, and let $\theta,\theta'\in\mathbb{S}^n$. The map $\Phi_{\mathbb{S}^n}$ is defined through the classical multidimensional scaling construction, and $d_{\mathbb{S}^n}$ denotes geodesic distance on the unit sphere. Spherical distance-map conjecture. For all $\theta,\theta'\in\mathbb{S}^n$,

$$\Vert \Phi_{\mathbb{S}^n}(\theta)-\Phi_{\mathbb{S}^n}(\theta')\Vert=\sqrt{\pi}\big(d_{\mathbb{S}^n}(\theta,\theta')\big)^{\frac{1}{2}}.$$

This conjecture was motivated by the exact calculation for $\mathbb{S}^1$ and by numerical computations; its validity for all higher-dimensional spheres remains open.