Dimension formula conjecture for Steinberg-generated cohomology

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 of $H^3(\Gamma,\mathbb{Q})$, and let $T$ be the Tits building for $\operatorname{GL}_3(\mathbb{Q})$. Dimension formula conjecture.

$$\dim_{\mathbb{Q}} H(\Gamma,E)=\dim_{\mathbb{Q}} H^3(\Gamma,\mathbb{Q})-\dim_{\mathbb{Q}} H_1(T/\Gamma,\mathbb{Q}).$$

The formula is presented as a numerical consequence of the preceding conjecture and was verified in the computations for the levels and fields tested, but remains open in the stated generality.