Let $D$ be a properly embedded free boundary disk in the closed unit ball $\mathbb{B}^3$ of $\mathbb{R}^3$. This means that $D$ is a smooth disk contained in this ball, $D \cap \partial \mathbb{B}^3 = \partial D$ and this intersection is orthogonal. By orthogonality here I mean this: if $N$ is a unit normal along $D$ (Gauss map), then $\langle N(x), x \rangle = 0$ for all $x \in \partial D$.
Assume that $D$ is strictly convex, that is to say, the principal curvatures are positive at each point of $D$ with respect to the fixed unit normal $N: D \to \mathbb{S}^2$. Does it follow that $N$ is a diffeomorphism onto its image? Equivalently, is $N$ injective?
The motivation is the following: if $S$ is a closed and connected surface in $\mathbb{R}^3$ which is also convex, then $N : S \to \mathbb{S}^2$ is a local diffeomorphism, hence a covering map. Since $\mathbb{S}^2$ is simply connected, this implies that $N$ is a global diffeomorphism. What happens when the surface has boundary?