The tempered local Langlands correspondence for rigid inner forms

Let $G^{*}$ be a connected, reductive, quasi-split group over the local function field $F$, with finite central $F$-subgroup $Z$, and let $G$ be an inner form of $G^{*}$. Fix a Whittaker datum $\mathfrak{w}$ for $G^{*}$. For a tempered Langlands parameter $\varphi\colon W_{F}'\to {}^{L}G$, set $S_{\varphi}=Z_{\widehat{G}}(\varphi)$ and let $S_{\varphi}^{+}$ be its preimage in $\widehat{\overline{G}}$, where $\overline{G}=G/Z$. Write $\Pi_{\mathrm{temp}}^{\mathrm{rig}}(G^{*})$ for the set of tempered irreducible admissible representations of rigid inner twists of $G^{*}$, and let $\operatorname{Irr}(\pi_{0}(S_{\varphi}^{+}))$ denote the irreducible representations of the finite group $\pi_{0}(S_{\varphi}^{+})$. The tempered local Langlands correspondence conjecture. There is a finite subset $\Pi_{\varphi}\subset\Pi_{\mathrm{temp}}^{\mathrm{rig}}(G^{*})$ and a commutative diagram

\begin{tikzcd} \Pi_{\varphi} \arrow["\iota_{\mathfrak{w}}"]{r} \arrow{d} & \operatorname{Irr}(\pi_{0}(S_{\varphi}^{+})) \arrow{d} \\ H^{1}(\mathcal{E}, Z \to G^{*}) \arrow{r} & \pi_{0}(Z(\widehat{\overline{G}})^{+})^{*} \end{tikzcd}

in which the top map is a bijection, the bottom map is given by the relevant pairing, the right map assigns to each irreducible representation the restriction of its central character to $\pi_{0}(Z(\widehat{\overline{G}})^{+})$, and the left map sends $(G_{1},\xi_{1},\mathscr{T}_{1},\bar h,\pi)$ to the class of $\mathscr{T}_{1}$. Moreover, there is a unique element $(G,\operatorname{id}_{G},G_{\mathcal{E}},\operatorname{id}_{\bar F},\pi)$ of $\Pi_{\varphi}$ such that $\pi$ is $\mathfrak{w}$-generic and $\iota_{\mathfrak{w}}$ identifies it with the trivial irreducible representation.