A pathological case of the $C_1$ conjecture in mixed characteristic

Let $K$ be a field of characteristic 0. Fix integers $r,d$ coprime with $r \geq 2$. Let $X_K$ be a smooth, projective, geometrically connected curve of genus $g \geq 2$ defined over K. Assume there exists a line bundle $L_K$ on $X_K$ of degree $d$. In this article we prove the existence of a stable locally free sheaf on $X_K$ with rank $r$ and determinant $L_K$. This trivially proves the $C_1$ conjecture in mixed characteristic for the moduli space of stable locally free sheaves of fixed rank and determinant over a smooth, projective curve.
Comment: Published in Mathematical Proceedings of the Cambridge Philosophical Society