Januszewski's polynomial multiplicity-growth conjecture

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 $\tau\in\hat{G}'$, write

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

Januszewski's conjecture. In the setting of Kobayashi's conjecture, $m_{\tau}(\pi|_{G'})$ grows at most polynomially in the norm of the infinitesimal character of $\tau$, in the sense of the source's multiplicity bound. This strengthens finite multiplicity to quantitative polynomial growth, but the source does not state that it has been proved.