Habegger's height conjecture for modularly related points

Let $C\subset \mathbb{A}^2$ be an irreducible algebraic curve defined over $\overline{\mathbb{Q}}$ that is not special. Let $s=(s_1,s_2)$ be a point of $C(\overline{\mathbb{Q}})$ satisfying $\Phi_M(s_1,s_2)=0$ for some positive integer $M$, where $\Phi_M$ is the relevant modular polynomial and $h$ is the logarithmic height. Habegger's height conjecture. There exists a positive constant $c$ such that, for all such points,

$$h(s)\leq c\max\{1,\log(M)\}.$$

This conjecture concerns the height of modularly related points on a non-special curve and is presented as a strengthening of the height bound used by Habegger and Pila. The source does not provide evidence of a resolution.