Generalized Hiraga–Ichino–Ikeda conjecture for tempered L-parameters

Let $G$ be a connected reductive group over $F$. For a tempered $L$-parameter $\phi$, let $\Pi_\phi(G)$ be its $L$-packet, let $\widetilde{\mathcal{S}}_\phi$ be the associated S-group, and let $\operatorname{Irr}(\widetilde{\mathcal{S}}_\phi,\zeta_G)$ denote the irreducible representations having central character compatible with the character $\zeta_G$ attached to the inner form $G$. Let $C_\phi'$ be the intersection of the centralizer $C_\phi$ with $\widehat{G/A}$. Generalized Hiraga–Ichino–Ikeda conjecture. There is a bijection

$$\Pi_\phi(G)\longrightarrow \operatorname{Irr}(\widetilde{\mathcal{S}}_\phi,\zeta_G)$$

such that, for every square-integrable representation $\pi\in\Pi_\phi(G)$,

$$\deg(\pi)=\zeta_\pi\frac{\dim\eta_\pi}{\#C_\phi'}\gamma(0,\pi,\operatorname{Ad},\psi),$$

where $\operatorname{Ad}:{}^LG\to\operatorname{GL}(\operatorname{Lie}(\widehat{G}_{\rm ad}))$ is the adjoint representation and

$$\zeta_\pi=\frac{|\gamma(0,\operatorname{St},\operatorname{Ad},\psi)|}{\gamma(0,\operatorname{St},\operatorname{Ad},\psi)}\frac{\epsilon(\frac12,\operatorname{St},\operatorname{Ad},\psi)}{\epsilon(\frac12,\pi,\operatorname{Ad},\psi)}\in\{\pm1\},$$

with $\operatorname{St}$ the Steinberg representation of $G(W)$. This conjecture refines the formal-degree conjecture by incorporating the internal parametrization of $L$-packets and a sign correction. The source does not provide evidence of resolution.