Sakellaridis–Venkatesh conjecture on parameters of relative discrete series

Let $X$ be a spherical variety for a reductive group $G$, let $G_X^\vee$ be its dual group, and let $\mathcal L_F$ denote the local Langlands group. An $A$-parameter $\psi: \mathcal L_F \times \operatorname{SL}(2,\mathbb C) \rightarrow G^\vee$ is $X$-distinguished if it factors through a distinguished morphism $\xi: G_X^\vee \times \operatorname{SL}(2,\mathbb C) \rightarrow G^\vee$, so that $\psi(w,g)=\xi(\psi_X(w),g)$ for a tempered $L$-parameter $\psi_X: \mathcal L_F\rightarrow G_X^\vee$. An $L$-parameter is elliptic when its image is not contained in any proper parabolic subgroup of $G^\vee$.

Sakellaridis–Venkatesh conjecture. A relative discrete series representation $\pi$ in $L^2(X)$ is contained in an Arthur packet corresponding to an $X$-distinguished $A$-parameter

$$\psi: \mathcal L_F \times \operatorname{SL}(2,\mathbb C) \rightarrow G^\vee$$

such that the $L$-parameter

$$\psi_X: \mathcal L_F \rightarrow G_X^\vee$$

is elliptic.

This conjecture describes the relative discrete spectrum of $X$ in terms of distinguished Arthur parameters and is part of the local conjectures of Sakellaridis and Venkatesh. The supplied context does not specify the extent to which this particular formulation has been proved.