Birational superrigidity and K-stability of projectively normal Fano manifolds of index one

We prove that every projectively normal Fano manifold in ${\mathbb{P}}^{n+r}$ of index $1$, codimension $r$ and dimension $n\ge 10r$ is birationally superrigid and K-stable. This result was previously proved by Zhuang under the complete intersection assumption.