Twisted GGP conjecture for unitary periods and twisted Asai L-values

Let $\pi$ be an irreducible cuspidal automorphic representation of $\mathrm U(V_{E_0})(\mathbb A_{E_0})$, let $\Pi$ be a weak base change of $\pi$ to $\mathrm {GL}_n(\mathbb A_E)$, and let $\omega=\omega(\mu,W)$ be the Weil representation for $V\otimes W$. Twisted GGP conjecture. The following are equivalent:

1. The period integral

$$\int_{[\mathrm U(V)]} \varphi(g)\overline{\phi(g)}\ne 0$$

for some $\varphi\in\pi$ and $\phi\in\omega$. 2. The twisted Asai L-value

$$L(1/2,\Pi,As_{E/F}\otimes\mu^{-1})\ne 0$$

and $\mathrm {Hom}_{\mathrm U(V)(\mathbb A)}(\varphi,\omega)\ne 0$. This is an unrefined conjectural relation between the nonvanishing of a twisted Fourier--Jacobi period and the corresponding central Asai value together with the occurrence of the Weil representation; its general status is not established in the supplied text.