Generalized multiplicity-one conjecture for $G_2$ Fourier-Jacobi models

Let $F$ be a $p$-adic field, let $\Pi$ be an irreducible selfdual representation of $G_2(F)$, let $\psi$ be a nontrivial additive character of $F$, and let $\widetilde\pi$ be an irreducible genuine representation of the double cover $\widetilde{\operatorname{SL}}_2(F)$. Generalized $G_2$ multiplicity-one conjecture. Then

$$\dim_J\bigl(\Pi,\widetilde\pi\otimes\omega_\psi\bigr)\leq1.$$

The conjecture is presented as an analogue of uniqueness of Fourier-Jacobi models for symplectic groups; the supplied text does not resolve it.