Voronoi-realized cohomology decomposition conjecture

Let $E$ be a real quadratic field and let $\Gamma\subset\operatorname{SL}_3(\mathbb{Z})$ be a finite-index subgroup. Let $H(\Gamma,E)$ be the Steinberg-generated subspace, and let $V$ denote its dual Voronoi realization in $H^3(\Gamma,\mathbb{Q})$. Let $H^3_!(\Gamma,\mathbb{Q})$ be the interior cohomology and let $A(\Gamma)$ be the subspace arising from cuspidal cohomology of the maximal Borel–Serre boundary faces. Voronoi-realized cohomology decomposition conjecture.

$$V(H(\Gamma,E))=H^3_!(\Gamma,\mathbb{Q})+A(\Gamma).$$

This is the paper's full conjecture, supported by the reported computations; its validity for arbitrary finite-index $\Gamma$ and real quadratic $E$ remains open.