The local rigid-inner-form packet conjecture

Let $F_v$ be a local completion of the global function field $F$, let $G$ be a connected reductive group over $F_v$, and let $W_{F_v}'$ be the Weil–Deligne group. For a tempered $L$-parameter

$$\varphi_v\colon W_{F_v}'\to\prescript{L}{}G,$$

write $\Pi_{\mathrm{temp}}^{\mathrm{rig}}(G)$ for the set of tempered representations of rigid inner forms of $G$, $S_{\varphi_v}^{+}$ for the relevant parameter centralizer, and $\mathcal{E}_v$ for the local gerbe defining rigid inner twists. The local rigid-inner-form packet conjecture. There is a finite subset $\Pi_{\varphi_v}\subset\Pi_{\mathrm{temp}}^{\mathrm{rig}}(G)$ and a commutative diagram

\begin{tikzcd} \Pi_{\varphi_v} \arrow["\iota_{\varphi_v,\mathfrak{w}_v}"]{r} \arrow{d} & \operatorname{Irr}(\pi_0(S_{\varphi_v}^{+})) \arrow{d} \\ H^{1}(\mathcal{E}_v,Z\to G^{*}) \arrow{r} & \pi_0(Z(\widehat{\overline{G}})^{+,v})^{*}, \end{tikzcd}

where $G^{*}$ is the quasi-split inner form and $Z$ is the chosen finite central subgroup. This is the predicted local Langlands-packet structure for rigid inner forms; the source presents it as part of the conjectural framework, and no resolution status is supplied.