The Steinberg-homology conjecture for real quadratic fields

Let $N$ be a level, let $E$ be a real quadratic field, and let $\psi_{\Gamma,E}$ denote the map associated with a subgroup $\Gamma$ as in the conjecture. Write $A_E(N)$ and $A_E(N)^*$ for the subgroups defined in the paper, and let $H_0^{\mathrm{cusp}}(\Gamma, \operatorname{St}(\mathbb{Q}^2;\mathbb{Z}))$ denote cuspidal degree-zero homology with coefficients in the Steinberg module. Conjecture. For any level $N$ and real quadratic field $E$: (1) if $\Gamma=\Gamma_1(N)^\pm$ or $\Gamma_1(N)$, then

$$\psi_{\Gamma,E}:\text{(source)}\longrightarrow H_0^{\mathrm{cusp}}(\Gamma, \operatorname{St}(\mathbb{Q}^2;\mathbb{Z}))$$

is surjective; (2) the cokernel of $\psi_{\Gamma_0(N)^\pm,E}$ is isomorphic to a quotient of

$$\bigl((\mathbb{Z}/N\mathbb{Z})^\times/\{\pm1\}\bigr)/A_E(N);$$

and (3) the cokernel of $\psi_{\Gamma_0(N),E}$ is isomorphic to a quotient of

$$(\mathbb{Z}/N\mathbb{Z})^\times/A_E(N)^*.$$

The conjecture extends the preceding computational and conditional results without assuming GRH. The source does not define the source of $\psi_{\Gamma,E}$ or the groups $A_E(N)$ and $A_E(N)^*$ in the supplied excerpt, so these definitions and the status of the conjecture should be checked against the full paper.