Frobenius descent conjecture for semi-stable bundles

Let $X$ be the genus-two curve in characteristic $2$, let $X_1$ be its Frobenius twist, let $F\colon X\to X_1$ be the relative Frobenius morphism, and let $\mathrm{M}_X$ and $\mathrm{M}_{X_1}$ denote the moduli spaces of semi-stable rank-$2$ vector bundles with the fixed determinant considered in the paper. Let $B$ be the bundle appearing in the condition below. Frobenius descent conjecture. For any semi-stable bundle $E\in \mathrm{M}_X$ satisfying

$$h^0(X,E\otimes B)=0,$$

there exists a semi-stable bundle $E_1\in \mathrm{M}_{X_1}$ such that

$$F^*E_1=E.$$

This conjecture asks whether every point outside the divisor defined by the vanishing condition admits a semi-stable Frobenius antecedent. It is presented as an optimistic conjecture concerning the surjectivity of the Frobenius pullback map; the supplied text gives no resolution.