Refined local Langlands compatibility conjecture for restriction and induction

Let $\overline{G}$ be a covering group with subgroup $\overline{G}_0$, let $\overline{G}^\dag=\overline{Z(G)}\cdot\overline{G}_0$, and let $\phi_\pi$ be an $L$-parameter of $\overline{G}$. Write $\phi_\pi^\diamondsuit=f_{G,H}\circ\phi_\pi$ and let $\Phi({}^L\overline{G}_0;\phi_\pi)$ be the set of parameters $\phi_0$ satisfying $f_{G_0,H}\circ\phi_0=\phi_\pi^\diamondsuit$. For enhanced parameters $(\phi,\theta)$, write $\mathcal{S}_\phi$ for the component group and use the embeddings into $\mathcal{S}_{\phi_\pi^\diamondsuit}$. Let $\pi\in\operatorname{Irr}_{\rm gen}(\overline{G})$ have central character $\omega_\pi$ and enhanced parameter $(\phi_\pi,\theta_\pi)$, and let $\tau\boxtimes\rho$ be an irreducible constituent of $\pi|_{\overline{G}^\dag}$. Restriction and induction compatibility conjecture. (i) The parameters satisfy $f_{G,z}\circ\phi_\pi=f_{c,z}\circ\phi_\tau=\phi_{\omega_\pi}$ and $\phi_\pi^\diamondsuit=f_{G_0,H}\circ\phi_\rho$. (ii) There exists $e(\pi)\in\mathbf{N}$ such that for every $\tau\in\operatorname{Irr}_{\rm gen}(\overline{G}_0)$ with enhanced parameter $(\phi_\tau,\theta_\tau)$,

$$\dim\operatorname{Hom}_{\overline{G}_0}(\pi,\tau)=\begin{cases}0&\text{if }\phi_\tau\notin\Phi({}^L\overline{G}_0;\phi_\pi),\\ e(\pi)\left\langle\operatorname{Ind}_{\mathcal{S}_{\phi_\pi}}^{\mathcal{S}_{\phi_\pi^\diamondsuit}}\theta_\pi,\operatorname{Ind}_{\mathcal{S}_{\phi_\tau}}^{\mathcal{S}_{\phi_\pi^\diamondsuit}}\theta_\tau\right\rangle_{\mathcal{S}_{\phi_\pi^\diamondsuit}}&\text{if }\phi_\tau\in\Phi({}^L\overline{G}_0;\phi_\pi).\end{cases}$$

In the isotypic case, the stated direct-sum formula for $\pi|_{\overline{G}_0}$ holds. (iii) Compatible $\tau\in\operatorname{Irr}_{\rm gen}(\overline{Z(G)})$ and $\rho\in\operatorname{Irr}_{\rm gen}(\overline{G}_0)$ induce constituents whose parameters fit into the specified commutative $L$-group diagram. This conjectural package predicts compatibility of enhanced local Langlands parameters with central characters, restriction, and induction.