SOLVED
Is there an entire function $f:\mathbb{C}\to \mathbb{C}$ such that, for any infinite sequence $n_1<n_2<\cdots$, the set
\[\{ z: f^{(n_k)}(z)=0 \textrm{ for some }k\geq 1\}\]
is everywhere dense?
Erdős
[Er82e] writes that this was solved in the affirmative 'more than ten years ago', but gives no reference or indication who solved it. From context he seems to attribute this to Barth and Schneider
[BaSc72], but this paper contains no such result.