Rigidity for proper holomorphic ball maps with degenerate CR Gauss map

Let $n$ and $N$ be integers with $2\leq n<N$, and let $F:\mathbb{B}^n\to\mathbb{B}^N$ be a proper holomorphic map that admits a $C^{N-n+1}$-smooth extension to the boundary. We prove that if the CR Gauss map of its boundary restriction is generically degenerate, then $F$ is holomorphically equivalen