Original Gross–Prasad conjecture for special orthogonal groups

Let $F$ be a number field with adèle ring $\mathbb{A}_F$, and let $V_n\subset V_{n+1}$ be quadratic spaces of dimensions $n$ and $n+1$ over $F$, with $n\geq 2$ and $V_n$ not a hyperbolic plane. Set $G_i=\operatorname{SO}(V_i)$, and let $\pi_n$ and $\pi_{n+1}$ be irreducible tempered cuspidal automorphic representations of $G_n(\mathbb{A}_F)$ and $G_{n+1}(\mathbb{A}_F)$, respectively. Assume that $\operatorname{Hom}_{G_n(F_v)}(\pi_{n+1,v}\otimes\pi_{n,v},\mathbb{C})\neq 0$ for every place $v$ of $F$. Original Gross–Prasad conjecture. There exist vectors $\varphi_i\in\pi_i$ such that

$$\int_{G_n(F)\backslash G_n(\mathbb{A}_F)}\varphi_{n+1}(g_n)\varphi_n(g_n)\,dg_n\neq 0$$

if and only if

$$L(1/2,\pi_{n+1}\boxtimes\pi_n)\neq 0.$$

Here the $L$-function is the product $L$-function. This conjecture relates the nonvanishing of a Gross–Prasad period integral to the central value of an automorphic product $L$-function; its resolution status is not specified in the source.