Ikeda's non-vanishing conjecture for Miyawaki liftings

Let $k$ be positive and even, let $f \in S_{2k-2(n+r)}(\operatorname{SL}_2(\mathbb{Z}))$ be a normalized Hecke eigenform, let $F^{(2n+2r)}$ be its Ikeda lift, and let $g \in S_k(\operatorname{Sp}_r(\mathbb{Z}))$ be a Hecke eigenform. Write $\mathcal{M}^{(2n+r)}(g,F^{(2n+2r)})$ for the Miyawaki lift. Ikeda's conjecture.

1. When $n=0$, the Miyawaki lift $\mathcal{M}^{(r)}(g,F^{(2r)})$ is nonzero if and only if the central value of $L(s,\operatorname{st}(g)\boxtimes f)$ is nonzero.
2. When $n>0$, the Miyawaki lift $\mathcal{M}^{(2n+r)}(g,F^{(2n+2r)})$ is always nonzero.

Ikeda proved that a nonzero Miyawaki lift has the expected standard $L$-function. The non-vanishing asserted here is still open and is the mechanism intended to establish Miyawaki's prediction.