Arithmetic Gan–Gross–Prasad conjecture in the Selmer rank one case

Assume the setup of the paper: $\Pi_0$ and $\Pi_1$ are the relevant representations under consideration, $F$ is the CM field, $E$ is their coefficient field, and the notation $\mathrm{V}_n$, $\Lambda_n$, $\mathrm{K}_n$, $\operatorname{Sh}$, $\phi_{\Pi_i}$, and the quotient Chow group is as in the statement. Arithmetic Gan–Gross–Prasad conjecture. Suppose that

$$L\left(\frac{1}{2},\Pi_0\times\Pi_1\right)=0\quad\text{and}\quad L'\left(\frac{1}{2},\Pi_0\times\Pi_1\right)\neq 0.$$

Then there exist a standard indefinite hermitian space $\mathrm{V}_n$ of rank $n$ over $F$, a self-dual lattice $\Lambda_n$, and an object $(\mathrm{K}_n,\mathrm{K}_{n+1})\in\mathfrak{K}(\mathrm{V}_n)_{\mathrm{sp}}$ with the specified local form, such that for every prime $\lambda$ of $E$, the graph $\triangle\operatorname{Sh}(\mathrm{V}_n,\mathrm{K}_n)$ of the morphism

$$\operatorname{Sh}(\mathrm{V}_n,\mathrm{K}_n)\longrightarrow\operatorname{Sh}(\mathrm{V}_{n+1},\mathrm{K}_{n+1})$$

is nonvanishing in the stated quotient Chow group. This is presented as a weak version of the arithmetic Gan–Gross–Prasad conjecture; the supplied passage gives no resolution status.