Vogan's local Langlands conjecture for tempered representations

Let $F$ be a non-archimedean local field, let $G^*$ be a quasi-split connected reductive group over $F$, and let $C$ be an algebraically closed field of characteristic $0$. For a tempered Langlands parameter $\varphi\in\Phi^{\rm temp}(G^*/F)$, let $S_\varphi^+$ be the inverse image of its centralizer in the universal cover of the dual group, and let $H^1_{\rm bas}(\mathcal{E}^{\rm rig},G^*)$ be the relevant basic cohomology set. Vogan's local Langlands conjecture. For each such $\varphi$ there should be a finite set $\Pi_\varphi$ of quadruples $(G,\xi,z,\pi')$ consisting of a rigid inner twist and a tempered irreducible representation, together with the stated commutative diagram whose top map is a bijection; its fibers over a fixed rigid inner twist should be the fibers of the local Langlands parametrization, and the packet should have a unique Whittaker-generic member for each Whittaker datum. In particular, the packet across inner twists is non-empty and finite. This refines the tempered local Langlands correspondence by incorporating inner forms, component-group representations, and Whittaker normalization; the general conjecture remains open.