Yoshida-type lifting conjecture for level 2 and level 4 elliptic forms

Let $S_{j,k}(\Gamma_2[2])$ be the space of Siegel cusp forms of weight $(j,k)$ on the principal level-$2$ subgroup, and let $w_2$ act on newforms of level $2$. For new eigenforms

$$f\in S_{l+m+4}(\Gamma_0(2))^{\rm new},\qquad g\in S_{l-m+2}(\Gamma_0(2))^{\rm new},$$

write $L(f,s)$ and $L(g,s)$ for their $L$-functions. Yoshida-type lifting conjecture. There is a Siegel Hecke eigenform $F\in S_{l-m,m+3}(\Gamma_2[2])$ with

$$L(F,s)=L(f,s)L(g,s-m-1).$$

It occurs with multiplicity $5$ when $f$ and $g$ have the same $w_2$-eigenvalue and with multiplicity $1$ when their eigenvalues are opposite. Likewise, for new eigenforms at level $4$,

$$f\in S_{l+m+4}(\Gamma_0(4))^{\rm new},\qquad g\in S_{l-m+2}(\Gamma_0(4))^{\rm new},$$

there is such an $F$ with the same spinor $L$-function, occurring with multiplicity $5$. This proposes a level-$2$ analogue of Yoshida lifting and explains predicted endoscopic pieces of the cohomology.