The spherical weight conjecture for the associated subgroup

Let $G/S$ be a spherical homogeneous space with associated dual group $\ch G$, let $\ch A^0$ be the torus introduced above, let $\cowts_{A^0}$ denote its relevant weight lattice, and let $\CV(G/S)$ be the valuation cone. For $\theta otin \cowts_{A^0}\cap \CV(G/S)$, write $'\IC^\theta$ for the corresponding irreducible object of $\catp_\CH({}'Z)$.

Spherical weight conjecture. For every $\theta\in \cowts_{A^0}\cap \CV(G/S)$, the irreducible object $'\IC^\theta\in \catp_\CH({}'Z)$ belongs to $\catq(Z)$.

This predicts that the category $\catq(Z)$ contains precisely the irreducible objects indexed by weights in the intersection of the relevant weight lattice with the valuation cone. The supplied text does not state whether the claim has been proved or disproved.