Ben-Zvi–Sakellaridis–Venkatesh period formula for spherical varieties

Assume that $\check{M}=T^*\check{X}$, where $\check{X}$ is a $\check{G}$-spherical variety. Let $\pi$ be a tempered, everywhere unramified automorphic representation of $G(\mathbb A)$ with $L$-parameter $\phi$. Suppose that $\phi$ has only finitely many fixed points $\{x_1,\ldots,x_r\}$ on $\check{X}$. For a suitably normalized spherical vector $f\in\pi$, Ben-Zvi–Sakellaridis–Venkatesh conjecture.

$$\mathcal{P}_M(f)\sim\sum_i L\bigl(0,(T_{x_i}\check{X})^{\mathrm{shear}}\bigr).$$

This is a proposed relation between automorphic periods and special values of $L$-functions in the relative Langlands program. The paper's abstract states that the formula is confirmed for certain cuspidal Eisenstein series, but the general statement presented here is not resolved by the supplied source.