The fibre-description conjecture for Frobenius pullback

Assume that $X$ is an ordinary smooth projective curve and that $V'$ is a general stable bundle of rank $r$. Let

$$\mathscr{V}^{-1}(V')\longrightarrow\mathscr{Q}^{r,0}_{V',0}$$

be the morphism sending $V$ to the inclusion $V\hookrightarrow F_*(V')$, where $\mathscr{Q}^{r,0}_{V',0}$ is the determinant-zero component of the Quot scheme. Fibre-description conjecture. This morphism is an isomorphism of smooth schemes. If true, the fibre of the Frobenius-pullback map can be identified with the indicated Quot-scheme component.