Horizontal diameter rigidity conjecture for singular Riemannian foliations on spheres

Let $\mathcal{F}$ be a singular Riemannian foliation on the unit sphere $\mathbb{S}^{n}$. A curve is horizontal if it meets the leaves of $\mathcal{F}$ perpendicularly, and $d_{\mathcal{H}}(p,q)$ denotes the infimum of the lengths of horizontal curves joining $p$ and $q$. Define the horizontal diameter by

$$diam_{\mathcal{H}}\mathbb{S}^{n}=\sup\{d_{\mathcal{H}}(p,q)\mid p,q\in\mathbb{S}^{n}\}.$$

Horizontal diameter rigidity conjecture. For any singular Riemannian foliation on a unit sphere $\mathbb{S}^{n}$, we have

$$diam_{\mathcal{H}}\mathbb{S}^{n}=\pi.$$

This asserts that the horizontal geometry of every singular Riemannian foliation on a round unit sphere has the same diameter as the sphere's antipodal distance. The result was proved for many classes of foliations, but the general case remains open in the supplied context.