Böcherer's Gross–Prasad type non-vanishing conjecture for Siegel cusp forms

Let $F \in S_k^{(2)}(N)$ be a Hecke eigenform, let $K$ be an imaginary quadratic field, and let $\Lambda$ be an ideal class group character of $K$. Suppose that the quantity $R(F,K,\Lambda)$ is nonzero. Böcherer's Gross–Prasad type conjecture. Then

$$L\left(\frac12,\pi_F\times\theta_\Lambda\right)\neq 0.$$

This conjecture is a generalization of a conjecture of Böcherer and has the form of a Gross–Prasad type non-vanishing prediction for Rankin–Selberg $L$-functions. It is open at the moment, although it has been proved for certain special Siegel cusp forms known as Yoshida lifts.