Stability of kernel bundles on projective bundles over curves

Let $X$ be a projective bundle over a smooth curve $C$ of genus $g \ge 3$, and consider the relative hyperplane bundle ${\mathcal O}_X (1) \to X$. Let $V \subseteq H^0 ( X , {\mathcal O}_X (1) )$ be a generating subspace. We prove that when $C$, $X$ and $V$ are general in moduli and ${\mathcal O}_X (1)$ is sufficiently ample, the kernel bundle of the system $({\mathcal O}_X (1) , V)$ is ${\mathcal O}_X (1)$-stable.