Let $C$ be a curve of genus $11$ or greater than or equal to $13$ in the primitive linear system of a K3 surface $X$ of Picard rank $1$. A recent theorem of Feyzbakhsh asserts that $X$ is (a Fourier-Mukai partner of) a Brill-Noether locus of higher rank vector bundles on $C$. In particular, $X$ is the unique K3 surface of Picard rank $1$ containing $C$. This proves a program originally suggested by Mukai. I will explain the argument for Feyzbakhsh’s theorem, based on wall-crossing techniques.