Rigidity for proper holomorphic ball maps with degenerate CR Gauss map

Let $n$ and $N$ be integers with $2\le n<N$, 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 equivalent, under composition with automorphisms of the source and target balls, to $V_m\oplus 0$ for some positive integer $m$, where $V_m(z)=(\sqrt{\frac{m!}{\alpha !}}{z}^{\alpha }{)}_{\mathrm{\mid }\alpha \mathrm{\mid }=m}$ is the degree-$m$ Veronese map. This resolves a problem posed by Xiaojun Huang.