Quantum modular invariant conjecture for real quadratic number fields

Let $\uptheta\in\mathbb R-\mathbb Q$ be a fundamental quadratic unit and let $K=\mathbb Q(\uptheta)$, with fundamental discriminant $D$. The multivalued quantum modular invariant $j^{\rm qt}(\uptheta)$ is defined using limits of approximants over diophantine approximations, and ${\sf N}^{\rm avg}(j^{\rm qt}(\uptheta))$ denotes the weighted product, or multiplicative expectation, of its values. Quantum modular invariant conjecture. There are $D$ distinct values of $j^{\rm qt}(\uptheta)$, all algebraic, and

$$H_K=K\bigl({\sf N}^{\rm avg}(j^{\rm qt}(\uptheta))\bigr),$$

where $H_K$ is the Hilbert class field of $K$. The conjecture extends the proposed solution of the Real Multiplication program from positive-characteristic global fields to real quadratic number fields; the observed finiteness, value patterns, and algebraicity of the quantum modular invariant remain unproved.