Spherical correlation conjecture

Let $\mathbb{S}^n$ be the canonical Riemannian sphere, let $o$ be the center of a fixed hemisphere, and write the hemisphere as $B(o,\pi/2)\subset\mathbb{S}^n$. Let $K_1,K_2$ be geodesically convex spherical bodies contained in this hemisphere and centrally symmetric around $o$. Spherical correlation conjecture.

$$\operatorname{vol}_n(K_1\cap K_2)\operatorname{vol}_n(B(o,\pi/2))\geq\operatorname{vol}_n(K_1)\operatorname{vol}_n(K_2).$$

This is the spherical analogue of the Gaussian and Cauchy correlation problems and is presented as a conjecture whose special cases are studied in the paper; the source does not state a resolution.