Finiteness conjecture for nonmodular orbifold Q-subcanonical Shimura curves

Let $X_\Gamma$ be a Shimura curve, and call it orbifold $\mathbb{Q}$-subcanonical when its orbifold canonical divisor has the corresponding $\mathbb{Q}$-subcanonical property.

Finiteness conjecture. There are only finitely many orbifold $\mathbb{Q}$-subcanonical Shimura curves, up to isomorphism, over $\mathbb{C}$ that are not modular curves.

This is motivated by the known finiteness of hyperelliptic Shimura curves; the claim concerns the nonmodular Shimura curves, which have no cusps.