Arthur-component-group conjecture for generic cuspidal representations

Let $G$ be a reductive group and let $\Psi^t(G)$ be the set of tempered Arthur parameters. For $\phi\in\Psi^t(G)$, let $\mathcal H_\phi^{\psi_N}$ be the nonzero irreducible generic constituent of the hypothetical Arthur-parameter space, denoted $\pi^{\psi_N}(\phi)$, and let $\mathcal S_\phi$ be the associated finite Arthur component group.

Arthur-component-group conjecture. For every $\phi\in\Psi^t(G)$,

$$c_{\pi^{\psi_N}(\phi)}^{\psi_N}=|\mathcal S_\phi|.$$

The conjecture identifies the Whittaker-Fourier coefficient with the size of the Arthur component group for every tempered parameter. The formulation depends on Arthur's conjectural decomposition and is open in the general number-field case.