Let $k$ be a perfect field, and let $\bar k$ be a fixed algebraic closure of $k$.
Let $\overline{X}$ be a nonempty smooth algebraic variety over $\bar k$.
Does there exist a natural number $d=d(\overline{X})$ with the following property:

> For any $k$-form $X$ of $\overline{X}$, the variety $X$ has a $K$-point over some finite field extension $K$ of $k$ of degree $[K:k]\le d$ ?

The answer YES would imply Theorem 2 of my answer to [this question](http://mathoverflow.net/questions/239022/).

[This question](http://mathoverflow.net/questions/128388/) and the references in comments to it may be relevant.