The Two Conjecture for quadratic Dirichlet characters

Let $\chi_1$ and $\chi_2$ be nontrivial, primitive Dirichlet characters modulo $q_1$ and $q_2$, respectively, with $\chi_1\chi_2(-1)=1$. Let $S_{\chi_1,\chi_2}$ be their generalized Dedekind sum, and let $\Gamma_1(q_1q_2)$ be the congruence subgroup appearing in its image. The Two Conjecture. If $\chi_1$ and $\chi_2$ are quadratic, then

$$S_{\chi_1,\chi_2}(\Gamma_1(q_1q_2))=2\mathbb{Z}.$$

The conjecture concerns the image of the newform Dedekind sum, which is known to be a full-rank lattice in the number field generated by the character values. The paper presents partial progress toward the containment in $2\mathbb{Z}$, but the equality remains open in the supplied source.