Almost multiplicity-one property of spherical varieties over finite fields

Let $H$ be a connected algebraic subgroup of a connected reductive group $G$ over a finite field ${\mathbb{F}}_q$ such that $G\mathrm{/}H$ is a $G$-spherical variety, i.e., $G\mathrm{/}H$ has an open dense $B$-orbit for each Borel subgroup $B$ of $G$. We formulate, for the pair $(G,H)$, an almost multiplicity-one property. Then we establish a criterion for this property in terms of the $B$-stabilizers on $G\mathrm{/}H$. In particular, we will see that this property is analogous to the strongly tempered condition in characteristic $0$.