Let $E$ be a rank $2$ stable vector bundle on a prime Fano threefold of genus $8$, with Chern numbers $c_1=1, c_2=6, c_3=0$.

Question.Is it true that $E(-1)=E^*$?

What I am able to show is that there is equality at the level of Chern characters, namely $\operatorname{ch}(E(-1))= \operatorname{ch}(E^*)$.