Lifting modular forms of half-integral weight to Siegel modular forms of even genus

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Pullback of the lifting of elliptic cusp forms and Miyawaki's conjecture (2006, background): “Note that Kohnen [20] defined a canonical linear map S k+(1/2)(Γ0(4)) → Sk(Sp2n+2r(Z)) h → F which coincides with the Duke-Imamoglu lifting when h is a Hecke eigenform.”
Siegel Modular Forms and Their Applications (2008, background): “He predicts in the case of genus g = 2 for even j ≥ 0 and k ≥ 3 an isomorphism S+j,k−1/2(Γ0(4), ψ) ∼−→S2k−6,j+3(Γ2) which should generalize the Shimura-Kohnen lifting S+k−1/2(Γ0(4)) ∼= S2k−2(Γ1), see [50].”
On the lifting of hermitian modular forms (2008, background): “Following Kohnen [14], we discuss the “linearization” of the lifting.”
The 1-2-3 of Modular Forms (2008, background): “He predicts in the case of genus g = 2 for even j ≥ 0 and k ≥ 3 an isomorphism S+j,k−1/2(Γ0(4), ψ) ∼−→S2k−6,j+3(Γ2) which should generalize the Shimura-Kohnen lifting S+k−1/2(Γ0(4)) ∼= S2k−2(Γ1), see [50].”