Multiplicity-one conjecture for the Schwartz space model

Let $P$ be a parabolic subgroup with Levi component $M$, let $X_P$ be the associated spherical variety, and set $\mathcal{S}:=\mathcal{S}(X_P)$. Let $\chi\boxtimes\pi\boxtimes\check{\pi}$ be an irreducible nice representation of $(M_\mathrm{ab}\times G\times G)(F)$. Vary $\chi$ in a family $\chi\otimes|\det|^\lambda$, with $\lambda\in\mathbb{C}$. Multiplicity-one conjecture. For $\chi$ in general position,

$$\dim \operatorname{Hom}_{M_\mathrm{ab}\times G\times G(F)}\left(\chi\boxtimes\pi\boxtimes\check{\pi},\mathcal{S}^\vee\right)\leq 1.$$

This is a uniqueness statement for the corresponding model and would support the expected local functional-equation formalism. The supplied text gives no resolution or evidence beyond presenting the assertion, so its status remains open.