Let $(G,H_1)$ and $(G,H_2)$ be spherical pairs (i.e. $G$ is a reductive group, $H_i$ are its closed subgroups and the Borel subgroup $B$ of $G$ has a finite number of orbits on $G/H_i$).

What can be said about the set of all elements $x \in G$ such that the conormal space at $x$ to its double coset $H_1xH_2$ does not intersect the nilpotent cone $\mathcal{N}$ in the dual Lie algebra of $G$? Is it open? Is it dense? Was it ever considered?

We know that if $H_1=H_2$ is a symmetric subgroup then this is the set of regular semisimple elements, which is open and dense.