Finiteness conjecture for good quaternionic Shimura levels

Fix an almost indefinite quaternion algebra $B$ over a totally real number field $F$. For an open compact subgroup $K$ of $B^\times(\mathbb A_{\mathrm f})$, let $X_K$ be the associated Shimura curve; for an open compact subgroup of $PB^\times(\mathbb A_{\mathrm f})$, let $Y_K$ be the corresponding projective Shimura curve. A curve is good if it satisfies the automorphic vanishing criterion for the modified diagonal cycle.

Finiteness conjecture. There are only finitely many open compact subgroups $K\subset B^\times(\mathbb A_{\mathrm f})$, respectively $K\subset PB^\times(\mathbb A_{\mathrm f})$, up to conjugation, such that $X_K$, respectively $Y_K$, is good and has genus greater than $0$.

This fixed-algebra finiteness statement is asserted to imply the broader conjecture on finiteness of good Shimura curves. The paper proves related finiteness results for quaternion algebras, but this level-by-level assertion remains open.