BZSV global numerical conjecture for periods on spherical varieties

Let $G$ be a reductive group, let $H$ be a subgroup equipped with a homomorphism $\boldsymbol{\rho}_H$ as in the BZSV setting, and let $\iota$ determine the dual homomorphism

$$\widehat{\iota}^{\prime}:\widehat{H}^{\prime}(\mathbb{C})\times \operatorname{SL}_2(\mathbb{C})\to\widehat{G}(\mathbb{C}).$$

For an irreducible automorphic representation $\pi$ of $G(\mathbb{A})$, let $\mathcal{P}_{H,\iota,\rho_H}(\varphi)$ denote the associated period integral, and let $\Pi$ be the relevant global Arthur parameter. BZSV global numerical conjecture. The period integral $\mathcal{P}_{H,\iota,\rho_H}(\varphi)$ is nonzero only if the Arthur parameter of $\pi$ factors through $\widehat{\iota}^{\prime}$. If this holds and $\pi$ is a lifting of a global tempered Arthur packet of $H^{\prime}(\mathbb{A})$, then

$$\frac{|\mathcal{P}_{H,\iota,\rho_H}(\varphi)|^2}{\langle\varphi,\varphi\rangle} “=" \frac{\displaystyle L\left(\frac{1}{2},\Pi,\rho_{\widehat{H}^{\prime}}\right)\cdot\prod_{k\in\widehat{I}}L\left(\frac{k}{2}+1,\Pi,\widehat{\rho}_k\right)}{L(1,\Pi,\operatorname{Ad})^2}.$$

Here $\langle\,,\,\rangle$ denotes a certain version of the $L^2$-norm. This is the numerical, or classical Langlands, form of the BZSV conjecture and predicts both the support of the period and its value in terms of special $L$-values. The displayed identity is stated conjecturally in the source; the paper verifies cases of the broader BZSV framework, but no resolution of this numerical conjecture is supplied here.