Effective André–Oort-type conjecture for pairs of almost holomorphic modular values

Let $V\subseteq\mathbb{C}^2$ be an algebraic curve defined over $\mathbb{Q}$. For quadratic points $\tau_1,\tau_2\in\mathbb{H}$, let $d_i$ denote the absolute value of the discriminant of $\tau_i$.

Effective André–Oort-type conjecture. There are effectively computable constants $c_i=c_i(V)$ such that whenever

$$(\chi^*(\tau_1),\chi^*(\tau_2))\in V,$$

either

$$\max(d_1,d_2)\leq c_1$$

or there is a primitive integer matrix $g$ of determinant at most $c_2$ such that

$$\tau_2=g\tau_1.$$

This is proposed as a more direct analogue of the effective André–Oort theorem for the modular invariant $j$. The statement is not proved in the source, and its notation is understood from the surrounding discussion of quadratic points and modular relations.