Sakellaridis–Venkatesh conjecture on Plancherel support

Let $F$ be a non-archimedean local field, let $\mathbf{G}$ be a reductive group over $F$, and let $\mathbf{X}$ be a $\mathbf{G}$-spherical $F$-variety. Write $G$ for the group of $F$-points of $\mathbf{G}$, let $G_X^\vee$ be the complex dual group associated with $X$, and let

$$\varrho:G_X^\vee\times \mathrm{SL}(2,\mathbb C)\longrightarrow G^\vee$$

be the distinguished morphism. An $A$-parameter is $X$-distinguished if it factors through $\varrho$ via a tempered $L$-parameter into $G_X^\vee$. Sakellaridis–Venkatesh's conjecture. The support of the Plancherel measure for $L^2(X)$, viewed as a representation of $G$, is contained in the union of Arthur packets attached to $X$-distinguished $A$-parameters. This is part of the conjectural description of harmonic analysis on spherical varieties; refined conjectures describe a direct-integral decomposition over these parameters, and the asserted containment is not established in the supplied source.