Let $S\subset\mathbb{P}^g$ be a polarised smooth projective K3 surface of genus $g$ (and degree $d=2g-2$) over $\mathbb{C}$. Denote by $\phi: S\to G(3,g+1)$ the Gauss map, taking a point $s\in S$ to its tangent 2-plane $\mathbb{T}_{S,s}\cong\mathbb{P}^2$ in $\mathbb{P}^g$. This is known to always be finite and birational, i.e. it is the normalisation onto its image.

Q:If $g$ is large enough and $S$ is general, is it true that $\phi$ is a closed embedding?

If $g=3$ then the Gauss map $\phi$ of a general quartic K3 $S\subset\mathbb{P}^3$ is ramified along the Hessian curve in $\mathcal{O}_S(8)$, on which it has degree 2, so in particular $\phi$ is not a closed embedding here.