On the orbit space of a maximal compact subgroup on a spherical homogeneous variety

Let $X=G\mathrm{/}H$ be a spherical homogeneous variety for a complex reductive algebraic group $G$. We prove that the orbit space of $X$ under the action of a maximal compact subgroup $K\subset G$ is homeomorphic to the valuation cone of $X$. We also discuss the relation between the orbit type stratification of the orbit space and the face stratification of the valuation cone.