Kobayashi's finite multiplicity conjecture for discretely decomposable restrictions

Let $(G,G')$ be a semisimple symmetric pair, let $\pi\in\hat{G}$ be an irreducible unitary representation of $G$, and suppose that the restriction of $\pi$ to $G'$ is infinitesimally discretely decomposable. For $\tau\in\hat{G}'$, the multiplicity is

$$m_{\tau}(\pi|_{G'})=\dim\operatorname{Hom}_{G'}(\tau,\pi|_{G'}).$$

Kobayashi's conjecture. The multiplicity $m_{\tau}(\pi|_{G'})$ is finite for every $\tau\in\hat{G}'$. This is a finite-multiplicity assertion for discretely decomposable restrictions; the source notes that little was known in general beyond Harish-Chandra's bound for multiplicities of $K$-types in finite-length representations.