Collinear triples conjecture for pairs of almost holomorphic modular values

Let $\chi^*$ be the almost holomorphic modular function appearing in the paper. For quadratic points $\tau_i\in\mathbb{H}$, define

$$P_1=(\chi^*(\tau_1),\chi^*(\tau_2)),\qquad P_2=(\chi^*(\tau_3),\chi^*(\tau_4)),\qquad P_3=(\chi^*(\tau_5),\chi^*(\tau_6)).$$

Collinear triples conjecture. There are only finitely many triples with the $P_i$ pairwise distinct and lying on a straight line which is neither horizontal, vertical nor the diagonal $x=y$.

This is proposed as a stronger analogue of the paper's finiteness theorem for collinear triples of points of the form $(j(\tau),\chi^*(\tau))$; the proposed statement remains unproved in the source.