Brill-Noether reconstruction of index one prime Fano threefolds

We show by a uniform argument that every index one prime Fano threefold $X$ of genus $g\geq 6$ can be reconstructed as a Brill-Noether locus inside a Bridgeland moduli space of stable objects in the Kuznetsov component $\mathcal{K}u(X)$. As an application, we verify Mukai's conjecture on the existence of dual embeddings of $X$. Moreover, we establish a refined categorical Torelli theorem for $X$ and classify autoequivalences of $\mathcal{K}u(X)$. We also give an alternative disproof of Kuznetsov's Fano threefold conjecture.