Let $Y$ be an affine open subset of a locally noetherian scheme $X$. Then, $X \setminus Y$ has pure codimension one [EGAIV$_4$, Cor. 21.12.7]. Moreover, if $X$ is proper and of finite type over a field $k$, and $\dim X \ge 2$, then $X \setminus Y$ is connected [Hartshorne, Cor. II.6.2].

My question is the following:

If $X$ is a complex projective variety of dimension $\ge 2$, and $Y$ is an open subset that is Stein, then is it still true that $X \setminus Y$ is connected?

The fact that $X \setminus Y$ is of pure codimension one is [Neeman, Prop. 3.4], and is a consequence of Hartog's theorem. Jing Zhang cites this proposition for the statement that $X \setminus Y$ is connected in Algebraic Stein Varieties, specifically in the proof of Thm. 2.8, but it is not clear to me how Neeman's proposition implies this result.

It could also be true that the algebraic proof when $Y$ is affine works in the case when $Y$ is Stein instead. But Hartshorne (and Goodman in his original article) use theorems about blow-ups from Nagata's paper on compactifications, and I don't know if these carry over to the analytic case.