Recall that a topological space $X$ is *scattered* if and only if every non-empty subset $Y$ of $X$ contains at least one point which is isolated in $Y$. Consider the statement: "Every scattered hereditarily separable space is countable".

In this book it is mentioned that, under $\mathsf{CH}$, K. Kunen constructed a compact, scattered, hereditarily separable space of size $\aleph_1$. My question is if a "real" example of a scattered, hereditarily separable, uncountable space is known, or if this statement is independent of $\mathsf{ZFC}$.