Finiteness conjecture for Q-subcanonical Hurwitz curves

A Hurwitz curve is a smooth projective curve attaining the Hurwitz automorphism bound. A point is $\mathbb{Q}$-subcanonical in the sense defined in the source.

Finiteness conjecture. There are only finitely many $\mathbb{Q}$-subcanonical Hurwitz curves.

The claim is proposed as an analogue, for Hurwitz curves, of the finiteness conjecture for nonmodular orbifold $\mathbb{Q}$-subcanonical Shimura curves. The source notes the relation between Hurwitz and Shimura curves but supplies no resolution.