Conjecture on multiplicity-free tempered restrictions and point reduced spaces

Let $G$ be a reductive group with maximal compact subgroup $K$. Let $H<G$ be a $\theta$-stable Cartan subgroup, and let

$$P=MAN<G$$

be a cuspidal parabolic subgroup corresponding to $H$, with $A$ the noncompact part of $H$. For each tempered representation $\pi$ induced from $P$, let $\Phi\colon G/H\to\mathfrak{k}^*$ be the corresponding moment map, and call its fibers modulo $K$ the reduced spaces. Multiplicity-free restriction conjecture. All tempered representations $\pi$ induced from $P$ restrict multiplicity-freely to $K$ if and only if all reduced spaces for all maps $\Phi$ corresponding to such representations are points.

This is stated as a partial converse to the preceding sufficient criterion for multiplicity-free restrictions. The supplied text does not provide a resolution of the equivalence.