The puncturing conjectures for specialness and the weak Hilbert property

Let $k$ be a field, let $X$ be a smooth variety over $k$, and let $Z\subset X$ be a closed subset of codimension at least two.

The puncturing conjectures. The following statements hold:

1. If $k=\mathbb{C}$ and $X$ is Brody-special, then $X\setminus Z$ is Brody-special.
2. If $X$ is geometrically-special over $k$, then $X\setminus Z$ is geometrically-special over $k$.
3. If $X$ is arithmetically-special over $k$, then $X\setminus Z$ is arithmetically-special over $k$.
4. If $X$ has the arithmetic weak Hilbert property over $k$, then $X\setminus Z$ has the arithmetic weak Hilbert property over $k$.

The corresponding puncturing statement for smooth special varieties is proved in the paper, and the Kobayashi-special case follows from classical invariance of the Kobayashi pseudometric. The Brody-special, geometric-special, arithmetic-special, and arithmetic weak Hilbert property cases are presented as conjectural.