Jacquet–Langlands correspondence for symplectic similitude groups

Let $S_k(N)$ be the space of Hilbert–Siegel cusp forms of weight $k$ and level $N$ on the split symplectic similitude group, let $S_k^B(N)$ be the corresponding space of algebraic Hilbert–Siegel cusp forms on the quaternionic inner form, and let $\mathbf{T}_k(N)$ and $\mathbf{T}_k^B(N)$ be their Hecke algebras away from $N$. Jacquet–Langlands correspondence. The Hecke algebras $\mathbf{T}_k(N)$ and $\mathbf{T}_k^B(N)$ are isomorphic, and there is a compatible isomorphism of Hecke modules

$$S_k(N)\stackrel{\sim}{\longrightarrow} S_k^B(N).$$

This conjecture predicts that the Hecke-module structure of the Hilbert–Siegel cusp forms on the split group can be computed through the quaternionic inner form, providing the proposed Jacquet–Langlands transfer for symplectic similitude groups. The supplied text does not state whether the correspondence has been proved or disproved.