Uniqueness conjecture for Fourier-Jacobi models of the split group of type G2

Let $k$ be a local field. Let $J={\mathrm{SL}}_2(k)\ltimes V$ be the subgroup of the split group ${\mathrm{G}}_2(k)$ described above, let $\widetilde{\mathrm{SL}}_2(k)$ be the metaplectic double cover of ${\mathrm{SL}}_2(k)$, and let $\omega_\psi$ be the corresponding Weil representation. For a self-dual irreducible representation $\sigma$ of ${\mathrm{G}}_2(k)$ and an irreducible genuine representation $\pi$ of $\widetilde{\mathrm{SL}}_2(k)$, regard $\pi\otimes\omega_\psi$ as a representation of $J$. Uniqueness conjecture. For any such $\sigma$ and $\pi$,

$$\dim\operatorname{Hom}_J(\sigma,\pi\otimes\omega_\psi)\leq 1.$$

This asserts multiplicity-one for the Fourier-Jacobi models under consideration, extending analogous uniqueness results for classical groups to the split exceptional group ${\mathrm{G}}_2$. The source proposes the statement for local fields but provides no resolution here.