Elliptic-orbit criterion for discrete-series infinitude on reductive homogeneous spaces

Let $G/H$ be a homogeneous space in the reductive setting, let $\mathfrak g$ and $\mathfrak h$ be the Lie algebras of $G$ and $H$, and let $\mathfrak h^{\perp}\subset\mathfrak g^*$ be the annihilator of $\mathfrak h$. Write $\mathfrak g_{\operatorname{ell}}^*$ for the elliptic part of $\mathfrak g^*$. Elliptic-orbit criterion. One has the equivalence

$$\#\operatorname{Disc}(G/H)=\infty \iff \mathfrak h^{\perp}\cap\mathfrak g_{\operatorname{ell}}^*\text{ contains a non-empty open set of }\mathfrak h^{\perp}.$$

This conjecture is a reformulation of the rank condition for reductive symmetric spaces and is known in that setting. The implication from the geometric condition to infinitude of the discrete series has been proved without assuming that $H$ is reductive; the converse remains open in the stated generality.