Miyawaki's conjecture for degree-three Siegel modular forms

Let $k$ be such that the indicated spaces are defined, and let $f \in S_{2k-4}(\operatorname{SL}_2(\mathbb{Z}))$ and $g \in S_k(\operatorname{SL}_2(\mathbb{Z}))$ be normalized Hecke eigenforms. Miyawaki's conjecture. There should exist a Hecke eigenform $F_{f,g} \in S_k(\operatorname{Sp}_3(\mathbb{Z}))$ whose standard $L$-function satisfies

$$L(s,F_{f,g},\operatorname{st})=L(s,g,\operatorname{Ad})L(s+k-2,f)L(s+k-3,f).$$

This predicts the existence of the Miyawaki lift attached to $f$ and $g$ and is reduced in the paper to a non-vanishing question for a Miyawaki lifting.