Minimalist Real Multiplication Values Conjecture

Prove that for any real quadratic irrational \(\rho\) with minimal polynomial \(a\rho^2 + b\rho + c = 0\) and non-square discriminant \(\Delta=b^2-4ac\), any rational vector \(\mathbf{r} \in \mathbb{Q}^2 \setminus \mathbb{Z}^2\), and any matrix \(A \in \Gamma_\mathbf{r}\) fixing \(\rho\), the special value \(\shin^\mathbf{r}_{A}(\rho)\) is algebraic, and that for any Galois automorphism \(g\) with \(g(\sqrt{\Delta}) = -\sqrt{\Delta}\), one has \(|g(\shin^\mathbf{r}_{A}(\rho))| = 1\).

Background

This conjecture provides a minimal algebraicity and unit-circle property for special values of the Shintani–Faddeev modular cocycle at real quadratic fixed points, sufficient (together with the Twisted Convolution Conjecture) to deduce SIC existence. It is strictly weaker than the Stark Conjecture and is implied by it according to the authors.