The Generalized Two Conjecture for quadratic number fields

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 $F_{\chi_1,\chi_2}$ be the smallest number field containing the values of $\chi_1$ and $\chi_2$, let $\mathbb{Z}[\chi_1,\chi_2]$ denote its ring of integers, and let $S_{\chi_1,\chi_2}$ be the generalized Dedekind sum. Generalized Two Conjecture. If $F_{\chi_1,\chi_2}$ is a quadratic number field, then

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

This generalizes the quadratic-character case by allowing the character-value field to be any quadratic number field. The claim is motivated by limited computations and is presented as an open generalization in the supplied source.