Non-vanishing conjecture for global Miyawaki liftings

Let $F$ be a totally real number field, let $\psi$ be a non-trivial unitary character of $\mathbb{A}/F$, and fix a totally positive $\xi\in F^\times$. Let $\pi$ be an irreducible representation of $\widetilde{\operatorname{Sp}}_r(\mathbb{A}_{\mathrm{fin}})$ occurring in the indicated holomorphic cuspidal space, and let $\mathcal{M}^{(n)}_{\psi,\tau}(\pi)$ be its Miyawaki lift. The global Miyawaki non-vanishing conjecture. Suppose that $n\geq r$.

1. If $n=r$, then $\mathcal{M}^{(r)}_{\psi,\tau}(\pi)$ is nonzero if and only if

$$L(1/2,\pi\times\tau\chi_{-1}^r)\ne0.$$

2. If $n>r$, then $\mathcal{M}^{(n)}_{\psi,\tau}(\pi)$ is always nonzero.

This extends part of Ikeda's conjecture. The simplest case over $\mathbb{Q}$ with $n=r=1$ and everywhere-unramified data is proved by Ichino and Xue, while examples in the opposite-parity case are known; the general assertion remains open.