Sakellaridis–Venkatesh weak local conjecture for spherical varieties

Let $Y=H\backslash G$ be a strongly tempered spherical variety over a local field $F$. Let $G_Y^\vee$ be the dual group of $Y$, let $f:G_Y^\vee\times\operatorname{SL}_2(\mathbb{C})\to G^\vee$ be the associated morphism, and let $\mu_{G_Y}^{\mathrm{PL}}$ be the Plancherel measure on $G_Y^\vee$-conjugacy classes of tempered Langlands parameters. A $Y$-distinguished Arthur parameter is a commutative diagram

\xymatrix{&G_Y^\vee\times\operatorname{SL}_2(\mathbb{C})\ar[dr]^f&\L_F\times\operatorname{SL}_2(\mathbb{C})\ar[rr]^\psi\ar[ur]^{(\phi,\operatorname{id})}&&G^\vee}

with $\psi$ an Arthur parameter and $\phi:L_F\to G_Y^\vee$ a tempered Langlands parameter. Weak local conjecture. There is a direct-integral decomposition

$$L^2(Y(F))\simeq\int_{[\psi]}\mathcal{H}_{[\psi]}\,d\mu_{G_Y}^{\mathrm{PL}}([\psi]),$$

where $[\psi]$ runs over $G_Y^\vee$-conjugacy classes of $Y$-distinguished Arthur parameters and $\mathcal{H}_{[\psi]}$ is isomorphic to a possibly empty direct sum of irreducible representations in the Arthur packet of $[\psi]$. This conjecturally describes the Plancherel decomposition of a spherical variety in terms of distinguished Arthur parameters; the source gives no resolution status.