Conjecture on algebraic independence of derivatives at quadratic points

Let $\mathcal{H}$ be the upper half-plane, and let $\tau_1,\dots,\tau_n\in\mathcal{H}$ be quadratic points lying in distinct $\operatorname{GL}_2^+(\mathbb{Q})$-orbits, with none lying in the $\operatorname{SL}_2(\mathbb{Z})$-orbit of $i$ or $\rho$. Algebraic-independence conjecture for derivatives at quadratic points. Then $j'(\tau_1),\dots,j'(\tau_n)$ are algebraically independent over $\overline{\mathbb{Q}}$. This conjecture is presented as the additional ingredient needed to obtain the preceding derivative-enhanced André–Oort statement in full generality; the supplied text gives no resolution.