Finiteness conjecture for congruence Fuchsian groups with vanishing modified diagonal cycles

Let $Gamma ot i {\rm SL}(2,\mathbb R)$ be a congruence Fuchsian group, and let $Gamma\backslash \mathbb H^*$ be the associated Shimura curve. A curve has vanishing modified diagonal cycles if its modified diagonal cycle is zero in the Chow group of its triple product.

Finiteness conjecture. There exist only finitely many congruence Fuchsian groups $\Gamma$ up to conjugation such that $\Gamma\backslash \mathbb H^*$ has vanishing modified diagonal cycles. Moreover, the set of almost indefinite quaternion algebras that can produce such $\Gamma$ is an effectively computable finite set.

This strengthens the finiteness theorem for congruence Shimura curves of bounded gonality, since subhyperelliptic curves have gonality at most $2$. The analogous assertion for arbitrary arithmetic groups is false by Belyi's theorem and the existence of hyperelliptic curves of arbitrarily large genus.