Let $X$ be a prime Fano threefold of index one and even genus $g\geq 6$, one can show that the moduli space of torsion free semistable sheaves $M(2,1,m_g)$ with $m_g=\left \lceil{\frac{g+2}{2}}\right \rceil$ consists of single point, this means that the only semistable torsion free sheaf $F$ with $\mathrm{ch}(F)=\mathcal{ch}(\mathcal{E})$ is $\mathcal{E}$, where $\mathcal{E}$ is the tautological sub-bundle on $X$ coming from the one in Grassmannian. If one assume that $F$ is a vector bundle, then the argument to prove the above statement is even easier. My question is that does this statement also hold of the semistable torison free sheaf $F$ with $\mathrm{ch}(F)=\mathrm{ch}(\mathcal{Q})$ such that $\mathcal{Q}$ is the tautological quotient bundle on $X$. If we assume $F$ is a vector bundle, it seems that I can figure out the proof by using Borel-Weil-Bott theorem on Grassmannian, but if I only assume that $F$ is a semistable torsion free sheaf, is this argument correct and how to argue it?
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