Recall that a cover $\mathscr{A}$ of a set $E$ is separating if for each pair of distinct points $p,q\in E$ there is $A\in \mathscr{A}$ such that $p\in A$ and $q\not\in A$. The separating weight of a $T_1$ space $X$, denoted $sw(X)$, is the smallest infinite cardinal $\kappa$ such that $X$ has a separating open cover $\mathscr{V}$ with $|\mathscr{V}|\leq \kappa$.
A basic fact about this cardinal function is the following: if $X$ is an infinite set, and $\tau$ and $\sigma$ are $T_1$ topologies on $X$ such that $\tau\subseteq \sigma$, then $sw(X,\sigma)\leq sw(X,\tau)$. From this we can deduce that, if $\tau_{cof}$ denotes the cofinite topology on $X$, and $\sigma$ is a $T_1$ topology on $X$, then the relations $sw(X,P(X)) \leq sw(X,\sigma) \leq sw(X,\tau_{cof})$ always hold.
Now, it is straightforward to show that $sw(X,\tau_{cof})=|X|$. My question is: can we determine the exact value of $sw(X,P(X))$?