Generic Beauville's Conjecture

Abstract: Let $\alpha: X \to Y$ be a finite cover of smooth curves. Beauville conjectured that the pushforward of a general vector bundle under $\alpha$ is semistable if the genus of $Y$ is at least $1$ and stable if the genus of $Y$ is at least $2$. We prove this conjecture if the map $\alpha$ is general in any component of the Hurwitz space of covers of an arbitrary smooth curve $Y$.