The ellipticity criterion for discrete series on real spherical spaces

Let ${\underline{G}}$ be a connected reductive group defined over $\mathbb{R}$, let $G={\underline{G}}(\mathbb{R})$, and let ${\underline{H}}\subset{\underline{G}}$ be an algebraic subgroup defined over $\mathbb{R}$, with $H={\underline{H}}(\mathbb{R})$. Set $Z=G/H$, and suppose that $Z$ is real spherical, meaning that a minimal parabolic subgroup $P$ of $G$ has an open orbit on $Z$. Let $\mathfrak{h}^{\perp}\subset\mathfrak{g}^*$ be the annihilator of $\mathfrak{h}$, and let $(\mathfrak{h}^{\perp})_{\rm ell}$ denote its elliptic subset. Ellipticity criterion. The real spherical space $Z$ admits discrete series representations if and only if the interior of $(\mathfrak{h}^{\perp})_{\rm ell}$ in $\mathfrak{h}^{\perp}$ is non-empty. This conjecture proposes a geometric criterion for the existence of discrete series representations on real spherical spaces; the paper explains that the criterion is known in the group case through Harish-Chandra's condition, while its validity in the general real spherical setting remains open.