Let $(G,H)$ be a spherical pair (i.e. $G$ is a reductive group, $H$ is a closed subgroup and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H$). Let $U$ be the unipotent radical of $B$.
What can be said about the set of all elements $x$ in $G/H$ such that the stabilizer of $x$ in $U$ is contained inside the derived group $[U,U]$?
Is it open? We know that it is a union of $B$-orbits. In the group case (i.e. $G=H \times H$ with $H$ diagonally embedded) we know that this is the open Bruhat cell. We do not know what it is even for general symmetric pairs.