Conjecture that semistability is equivalent to cohomological semistability for vector bundles on curves

Let $C$ be a smooth projective curve defined over an algebraically closed field $k$ of characteristic $p>0$, and let $E$ be a vector bundle over $C$.

Semistability–cohomological semistability conjecture. $E$ is semistable if and only if $E$ is cohomologically semistable.

In characteristic zero, exterior powers preserve semistability, so cohomological semistability is equivalent to slope semistability. In positive characteristic, exterior powers need not preserve semistability, and the preceding proposition shows that stability is not equivalent to cohomological stability; the conjecture asks whether the corresponding equivalence nevertheless holds for semistability.