Batyrev's conjecture on K-orbits and the valuation cone

Let $G/H$ be a spherical homogeneous space, let $K$ be a maximal compact subgroup of $G$, and let $\mathcal{V}_{G/H}$ be the valuation cone. For each face $\sigma$ of $\mathcal{V}_{G/H}$, let $X_\sigma$ denote the corresponding boundary face of the stratification of $K\backslash G/H$. Batyrev's conjecture. The space $K\backslash G/H$ is a stratified manifold with corners whose boundary faces $X_\sigma$ are in natural bijection with the faces $\sigma$ of $\mathcal{V}_{G/H}$; the stabilizers of points in the relative interior of $X_\sigma$ are maximal compact subgroups in the corresponding satellite subgroup of $H$. Moreover, there is a homeomorphism of stratified spaces

$$\Phi:\mathcal{V}_{G/H}\to K\backslash G/H$$

that restricts, for every face $\sigma\subset\mathcal{V}_{G/H}$, to a homeomorphism between the relative interiors of $\sigma$ and $X_\sigma$. The conjecture proposes a precise relation between the compact orbit space and the valuation cone, supported in the paper by examples but not established in general.