Arthur-parameter conjecture for Whittaker-Fourier coefficients

Let $G$ be a reductive group, let $\rho_N$ be a fixed Whittaker datum, and let $\rho_N$-generic cuspidal representations be measured by their Whittaker-Fourier coefficients. For an elliptic Arthur parameter $\rho$ of Ramanujan type, let $\rho^{\rho_N}(\rho)$ denote the associated irreducible generic cuspidal representation, and let $\S_\rho$ be its Arthur component group.

Arthur-parameter conjecture. For every elliptic Arthur parameter $\rho$ of Ramanujan type,

$$c_{\rho^{\rho_N}(\rho)}^{\rho_N}=|S_\rho|.$$

This predicts that the relevant Whittaker-Fourier coefficient is governed by the size of Arthur's component group. It relies on the conjectural decomposition into Arthur-parameter spaces and remains open in the general number-field setting.