Generic Beauville’s Conjecture

Let $\alpha \colon X \to Y$ 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 .