The finiteness is known for any scheme $Y$ which is regular, flat and proper over Spec Z. This is due to the work of Bloch, Kato-Saito among others; see [Szamuely's Seminaire Bourbaki expose][1] for a modern account. Your $X$ may be an open subscheme of a $Y$; since the fibers of $X$ (and $Y$) over Spec Z are one-dimensional, the finiteness for $Y$ implies that for $X$.


  [1]: http://www.renyi.hu/~szamuely/bour09.pdf