Finiteness conjecture for B-modular curves

Let $F$ be a field, let $B$ be a quaternion algebra over $F$, and let $g>1$. A curve $C/F$ is $B$-modular if there is a Shimura curve $X_K$ associated with a suitable compact open subgroup $K\subset B^\times({\mathbb A}_{\mathrm f})$ and a non-constant morphism $X_K\to C$. Finiteness conjecture for $B$-modular curves. Given $B$ and $g>1$, there are only finitely many $B$-modular curves of genus $g$. This generalizes the cited conjecture for the split quaternion algebra and predicts finiteness of modular curves of each fixed genus; no resolution is given in the supplied text.