The genus-two Rankin-product lifting conjecture to genus four

Let $f$ and $g$ be Siegel modular forms of genus $2$, of weights $k>4$ and $l=k-2$, with Satake parameters $(\alpha_0,\alpha_1,\alpha_2)$ and $(\beta_0,\beta_1,\beta_2)$, respectively. Genus-two Rankin-product lifting conjecture. There exists a Siegel modular form $F$ of genus $4$ and weight $k$ whose Satake parameters can be chosen as

$$\gamma_0=\alpha_0\beta_0,\quad \gamma_1=\alpha_1,\quad \gamma_2=\alpha_2,\quad \gamma_3=\beta_1,\quad \gamma_4=\beta_2.$$

The conjecture is motivated by an equality of Hecke-series denominators and by examples from Ikeda–Miyawaki constructions, but the source does not claim a proof in general.