Hiraga–Ichino–Ikeda–Langlands conjecture for Plancherel densities

Let $G$ be the reductive group under consideration, let $\operatorname{Temp}(G)$ be its tempered dual, and let $\pi\in\operatorname{Temp}(G)$. Let $\rho_{\pi,\varpi}$ be the representation of the component group $S_{\phi_\pi}$ associated with the $L$-packet parameterization, and let $\gamma(s,\pi,\operatorname{Ad}_G,\psi)$ be the adjoint gamma factor. If $n_\pi$ is the order of the zero of this gamma factor at $s=0$, define

$$\gamma^*(0,\pi,\operatorname{Ad}_G,\psi):=\left(\zeta_F(s)^{n_\pi}\gamma(s,\pi,\operatorname{Ad}_G,\psi)\right)_{s=0},$$

where $\zeta_F(s)$ is the local zeta factor of $F$. Hiraga–Ichino–Ikeda–Langlands conjecture. For almost all $\pi\in\operatorname{Temp}(G)$,

$$\mu_G(\pi)=\frac{\deg(\rho_{\pi,\varpi})}{\lvert S_{\phi_\pi}\rvert}\left\lvert\gamma^*(0,\pi,\operatorname{Ad}_G,\psi)\right\rvert.$$

This prediction combines the conjectural Langlands normalization of intertwining operators with the formal-degree conjecture to describe Plancherel densities. Its status is not specified in the supplied text.