Paramodular eigenform contribution to cuspidal cohomology of \SL_4$\mathbb Z$

Let $N$ be prime, and let $P_3^{\mathrm{nG}}(N)$ denote the space of weight-three non-Gritsenko paramodular eigenforms. Choose an equivalence class of eigenforms in $P_3^{\mathrm{nG}}(N)$ and a representative $h$. Let $d_h$ be the degree of the extension of $\mathbb Q$ generated by the eigenvalues of $h$. Let $H^5_{\mathrm{cusp}}(\Gamma_0(N);\mathbb C)$ be the cuspidal fifth cohomology, and let $H(\xi)$ and $H_{\operatorname{Sp}}(h)$ denote the Hecke polynomials defined in the paper.

Paramodular cohomology conjecture. The cuspidal cohomology $H^5_{\mathrm{cusp}}(\Gamma_0(N);\mathbb C)$ contains a $2d_h$-dimensional subspace spanned by Hecke eigenclasses. For every eigenclass $\xi$ in this subspace, the Hecke polynomial $H(\xi)$ agrees with $H_{\operatorname{Sp}}(h)$ up to Galois conjugacy.

This predicts the contribution of non-Gritsenko paramodular forms to the cuspidal cohomology at prime level. The supplied text gives no resolution evidence, so the conjecture is recorded as open.