Let $X$ be a Banach space. Is the following implication valid?

$$ (X,w) \textrm{ is hereditarily Lindelöf}~ \Rightarrow X^*~ \textrm{is separable} $$

The converse is clearly true, since the closed unit ball is relatively weak star second countable.

**Def.** A topological space $X$ is hereditarily Lindelöf if every subspace $Y\subseteq X$ is Lindelöf.