Description of the strong approximation locus using Brauer-Manin obstruction for homogeneous spaces with commutative stabilizers

For a homogeneous space X over a number field k, the Brauer-Manin obstruction has been used to study strong approximation for X away from a finite set S of places, and known results state that $X(k)$ is dense in the omitting-S projection of the Brauer-Manin set prS(X(Aₖ)ᵇʳ), under certain assumptions. In order to completely understand the closure of $X(k)$ in the set of S-adelic points X(AₖS), we ask: (i) whether prS(X(Aₖ)ᵇʳ) is closed in X(AₖS); (ii) whether $X(k)$ is dense in the closed subset of X(AₖS) cut out by elements in brX which induce zero evaluation maps at all the places in S. We also ask these questions considering only the algebraic Brauer group. We give answers to such questions for homogeneous spaces X under semisimple simply connected groups with commutative stabilizers.