Duflo's branching and moment-map conjecture for discrete series

Let $G$ be a real almost algebraic group, let $\pi$ be a discrete series representation attached to a coadjoint orbit $\mathcal{O}_{\pi}$, and let $H$ be a closed almost algebraic subgroup. Let

$$q\colon \mathcal{O}_{\pi}\rightarrow \mathfrak{h}^{\ast}$$

be the moment map. Duflo's conjecture. (i) The restriction $\pi\vert_{H}$ is $H$-admissible if and only if $q$ is weakly proper. (ii) If $\pi\vert_{H}$ is $H$-admissible, every irreducible $H$-representation $\sigma$ appearing in it is attached to a strongly regular $H$-coadjoint orbit $\mathcal{O}'_{\sigma}$ contained in $q(\mathcal{O}_{\pi})$. (iii) If $\pi\vert_{H}$ is $H$-admissible, the multiplicity of each such $\sigma$ can be expressed geometrically in terms of the reduced space $q^{-1}(\mathcal{O}'_{\sigma})/H$. This conjecture connects branching laws with the geometry of moment maps and is motivated by the orbit method and quantization-commutes-with-reduction. The paper verifies it for $G=\operatorname{Spin}(N,1)$ and parabolic subgroups, while the general statement remains open.