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Let $K$ be a field, $G$ a smooth finite linear algebraic group over $K$, $X$ a proper reduced connected separated scheme of finite type over $K$, $g: Y \to X$ a connected etale $G$-torsor over $X$ (so in particular $g$ is finite surjective and etale). Is the following statement true

For any (Zariski) open dense $U \subset X$ there exists an open dense $V \subset X$ contained in $U$ such that $g^{-1}(V)$ is connected.

If not, what additional conditions can we put on $X$ so that it becomes true?

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    $\begingroup$ This is automatic if $X$ is normal (or more generally geometrically unibranch), for then so is $Y$, so any open in $X$ or $Y$ is irreducible if and only if it is connected, and a nonempty open in an irreducible is irreducible. To see that it fails already for nodal curves, look at Figure 12 in Exercise III.10.6 in Hartshorne (take $U$ the complement of the node). $\endgroup$
    – R. van Dobben de Bruyn
    Commented Nov 10, 2019 at 4:58

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