Finiteness of the number of classes of vector bundles on $\mathbb{P}^1_{\mathbb{Z}}$ with jumps of height $2$

We consider vector bundles of rank $2$ with jumps of heights $1$ and $2$ and a trivial generic fiber on the arithmetic surface $\mathbb{P}^1_{\mathbb{Z}}$. The finiteness of the number of isomorphism classes of such vector bundles with a fixed discriminant and, as a consequence, with a fixed genus is obtained.