Bounds on multiplicities of spherical spaces over finite fields -- the general case

Let $G$ be a connected reductive group scheme acting on a spherical scheme $X$. In the case where $G$ is of type $A_n$, Aizenbud and Avni proved the existence of a number $C$ such that the multiplicity $\dim \mathrm{hom}(\rho ,\mathbb{C}[X(F)])$ is bounded by $C$, for any finite field $F$ and any irreducible representation $\rho$ of $G(F)$. In this paper, we generalize this result to the case where $G$ is a connected reductive group scheme over $\mathbb{Z}$, and prove Conjecture A of [1].