The illumination conjecture for modular curves

Let $X(d)$ be the modular curve equipped with its square-tiling and the metric induced by the quadratic differential $q$. Say that a point $A\in X(d)$ illuminates a point $B\in X(d)$ if a geodesic segment for the metric $|q|$ connects $A$ to $B$ without passing through a singularity. Illumination conjecture. The cusps of the modular curve illuminate all of $X(d)$ except possibly some vertices of the square-tiling. This conjecture reformulates one aspect of the parity conjecture in terms of geodesic illumination on $X(d)$; the supplied text does not state a resolution.