Miyawaki–Ikeda lifting conjecture

Let $k$ and $n$ be natural numbers with $k-n$ even. Let $f\in S_{2k}(\Gamma_1)$ be a normalized Hecke eigenform, and let $F_{2n}\in S_{k+n}(\Gamma_{2n})$ be the Ikeda lift of $f$. Miyawaki–Ikeda conjecture. For every eigenform $g\in S_{k+n+r}(\Gamma_r)$ with $n,r\geq 1$, there exists a Siegel modular eigenform ${\mathcal F}_{f,g}\in S_{k+n+r}(\Gamma_{2n+r})$ such that

$$D_{{\mathcal F}_{f,g}}(s)=Z_g(s)\,\prod_{j=1}^{2n}L_f(s+k+n-j),$$

where $L_f=Z_f$ is the usual $L$-function. This conjectures the existence of a Miyawaki–Ikeda lift with the specified standard $L$-function factorization, extending the lifting phenomena discussed earlier; the source gives no resolution status.