Let $f_w:\mathbb C \to \mathbb C$ be an entire function such that $(0,1) \ni w \mapsto f_w$ is real-analytic.

Assuming that there is a dense subset $D \subset (0,1)$ such that for $w \in D$ the function $f_w$ has infinitely many zeros. What does this imply for the number of zeros of $f_w$ with $w\in(0,1) \setminus D?$

In a previous question I already learned that I cannot expect $f_w$ to have always infinitely many zeros as well.

This happened in this question.

I wonder whether $f_w$ has to have infinitely many zeros

1.) for some $w \in (0,1) \setminus D?$

2.) for almost all $w \in (0,1)$ in the Lebesgue sense?

3.) for generic $w \in (0,1)$ in the Baire sense?

4.) for all $w\in (0,1)$ aside from finitely many?