Mukai bundles and Brill-Noether generality for prime Fano threefolds in positive characteristic

Let $X$ be a prime Fano threefold of genus $g\ge 8$ in positive characteristic. For every nontrivial decomposition $g=rs$, we prove that there exists a unique Mukai bundle on $X$ of type $(r,s)$. To this end, we show that every smooth hyperplane section of $X$ is Brill-Noether general.