Refined local L-packet conjecture for unitary similitude groups

Let $F$ be a local field, let $G=\operatorname{GU}_n$, let $\RU_n$ be the associated unitary group, and let $\phi\in\Phi_{\mathrm{bdd}}(G)$ be a bounded Langlands parameter. Let $\widetilde{\zeta}$ be the chosen central character, let $\widetilde{\Pi}_{\phi,\widetilde{\zeta}}$ be the corresponding coarse packet, let $X$ be the group of twisting characters, and let $\alpha(\mathcal S_\phi)$ be the image of the component group under the twisting map. For a test function $\widetilde f\in C_c^\infty(\operatorname{GU}_n(F),\widetilde{\zeta}^{-1})$, write $\widetilde f_{\operatorname{GU}_n}(\widetilde\pi)$ for the character distribution of $\widetilde\pi$. Refined local $L$-packet conjecture. There exists a subset $\Pi_{\widetilde\phi}$ of $\widetilde\Pi_{\phi,\widetilde\zeta}$, unique up to twisting by $X$, such that

$$\widetilde{\Pi}_{\phi,\widetilde{\zeta}}=\bigsqcup_{\omega\in X/\alpha(\mathcal S_\phi)}\Pi_{\widetilde\phi}\otimes\omega,$$

and such that

$$\widetilde f(\widetilde\phi):=\sum_{\widetilde\pi\in\Pi_{\widetilde\phi}}\widetilde f_{\operatorname{GU}_n}(\widetilde\pi)$$

is stable for every such test function. Here $C_c^\infty(\operatorname{GU}_n(F),\widetilde\zeta^{-1})$ is the space of $\widetilde\zeta$-equivariant smooth functions with compact support modulo $Z_{\operatorname{GU}_n}(F)$, and

$$\widetilde f_{\operatorname{GU}_n}(\widetilde\pi)=\operatorname{tr}\pi(\widetilde f)=\operatorname{tr}\int_{Z_{\operatorname{GU}_n}(F)\backslash\operatorname{GU}_n(F)}\widetilde f(x)\pi(x)\\,dx.$$

This conjectural refinement is intended to provide a section of the pairing between coarse packets and component groups; the source relates it to prior work of Xu. Its general validity is not established in the supplied text.