Metaplectic tensor product and restriction parameter conjecture

Let $\overline{M}_{\mathbf r}$ be the covering Levi subgroup with dual inclusion $i_M:\overline{M}_{\mathbf r}^\vee\hookrightarrow\prod_{i=1}^k\overline{{\rm GL}}_{r_i}^\vee\times\overline{{\rm GSp}}_{2r_0}^\vee\times\mathbf{C}^\times$. Let $\pi_i$ and $\pi_0$ have parameters $\phi_i$ and $\phi_0$, and let $\phi_\omega$ be the relevant central-character parameter. Metaplectic parameter compatibility conjecture. (i) The metaplectic tensor product $\widetilde{\otimes}_i\pi_i$ has parameter exactly $(\phi_1\times\cdots\times\phi_k\times\phi_0)\times\phi_\omega$, which factors through $i_M$ and hence takes values in $\overline{M}_{\mathbf r}^\vee$. (ii) The metaplectic restriction $\widetilde{\rm R}(\Pi)$ has parameter exactly $i_M\circ\phi_\Pi$. This predicts compatibility of the metaplectic constructions with the local Langlands correspondence.