Multiplicity-one conjecture for the $G_2$ Fourier-Jacobi model

Let $F$ be a $p$-adic field, let $\Pi$ be an irreducible generic representation of $G_2(F)$, let $\psi$ be a nontrivial additive character of $F$, and let $\widetilde I(\chi,\psi)$ be the genuine induced representation on the double cover $\widetilde{\operatorname{SL}}_2(F)$ associated with a character $\chi$ of $F^\times$. $G_2$ multiplicity-one conjecture. If $\widetilde I(\chi,\psi)$ is irreducible, then

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

This minimal multiplicity-one statement is proposed to provide the local functional equation for Ginzburg's local zeta integral and hence the $\operatorname{GL}_1$-twisted gamma factors; its status is not resolved in the supplied text.