The finite-fibre conjecture for Frobenius pullback

Assume that $X$ is an ordinary smooth projective curve and let $V'$ be a general stable bundle of rank $r$ and trivial determinant. Let $F$ denote the absolute Frobenius and define

$$\mathscr{Q}^{r,0}_{V'}=\operatorname{Quot}^{r,0}(F_*(V')).$$

Finite-fibre conjecture. The quotient scheme $\mathscr{Q}^{r,0}_{V'}$ is finite and reduced. This conjecture is motivated by analogous characteristic-zero results; it is presented as conjectural in the supplied text.