Is every semi-stratifiable space $\omega$-monolithic?

Definitions

A topological space $(X,\tau)$ is called semi-stratifiable if there exists a function $g:\omega\times X\to\tau$ such that:

1. for any point $x$ of $X$ holds $\{x\}=\bigcap_{n\in\omega} g(n,x)$;

2. for any point $x$ of $X$ and a sequence $\{x_n\}$ of $X$, if $x \in g(n,x_n)$ for each $n$, then $x_n \to x$.

A topological space $X$ is said to be $\omega$-monolithic if $nw(\overline{A}) \le \omega$ for any subset $A \subset X$ with $|A| \le \omega$.