Yoshida-lift generation conjecture for vector-valued Siegel modular forms

Let $S_{j,2}(\Gamma_2[2])$ denote the space of vector-valued Siegel modular forms of degree $2$, weight $(j,2)$, and level $\Gamma_2[2]$. A Yoshida-lift generation conjecture. The space $S_{j,2}(\Gamma_2[2])$ is generated by Yoshida type lifts. The preceding theorem describes the relevant Yoshida lifts and their multiplicities in the cases where they occur; the conjecture asserts that these lifts account for the whole space.