Let $X$ be a Banach space. It is natural for us to introduce a quantity measuring the separability of sets as follows: for a subset $A$ of $X$, we set

$\textrm{sep}(A)=\inf\{\epsilon>0: A\subseteq K+\epsilon B_{X}$ for some countable subset $K$ of $X\}$.

Clearly, $A$ is separable if and only if $\textrm{sep}(A)=0$.

It is elementary that a Banach space $X$ is separable if $X^{*}$ is separable. My question is about a quantitative version of this result.

Question.Does there exist a universal constant $C$ such that $$\textrm{sep}(B_{X})\leq C\cdot \textrm{sep}(B_{X^{*}}) \, ?$$