Kobayashi's discrete-series infinitude conjecture for reductive homogeneous spaces

Let $G/H$ be a reductive homogeneous space in Setting, and let $\operatorname{Disc}(G/H)$ denote its discrete series representations. Kobayashi's conjecture. One has the equivalence

$$\operatorname{Disc}(G/H) \ne \emptyset \iff \# \operatorname{Disc}(G/H)=\infty.$$

This conjecture would help characterize reductive homogeneous spaces admitting discrete series representations. It is true for reductive symmetric spaces, and the database status is solved because the cited source states that the relevant implication has been proved without assuming that $H$ is reductive.