Intersection of dense sets in a Banach space

Suppose $(X, ||.||_X)$ be a Banach space. Let $Y, Z$ be two dense subsets.

We know that $Y \cap Z$ may not be dense in $X$ (rationals and irrationals on real line) even if $Y \cap Z \neq \emptyset$ (rationals $\cup \sqrt{2}$ and irrationals on real line).

But if $Y \cap Z$ contains an open subset of $(X, ||.||_X)$, can we say $Y \cap Z$ is dense? No: we can take $X=\mathbb{R}$ with Euclidean norm, $Y=\mathbb{Q} \cup (0,2)$ and $Z=(\mathbb{R} \setminus \mathbb{Q}) \cup (0,2)$.

From the links below we can see,

it is a sufficient criterion for $Y \cap Z$ to be a dense subset of $(X, ||.||_X)$ if either of $Y$ or $Z$ is open in $X$. But is it also a necessary criterion ?

Any reference to the proof of this fact ?

Some related links can be found below:

1. Dense subset of two Banach spaces also dense in the intersection

2. In a complete metric space with no isolated points, any countable intersection of open dense sets is uncountable?

3. Intersection of an open dense set and any dense set is dense.

4. If $A, B$ are open dense subsets of a metric space $X$, is their intersection dense??