Let $(X,\mu,\sigma)$ be a subshift on a finite alphabet, which we assume to be mixing. That is, for any cylinders $C, C'$ we have $\mu(\sigma^{-n}C\cap C')\to\mu(C)\mu(C')$ as $n\to+\infty$. We also assume $h_\mu(\sigma)$, the metric entropy of our subshift, to be positive.

Let $F_n$ denote the set of all cylinders of the form $[x_1=i_1,\dots,x_n=i_n]\subset X$.

**Question.** Is there a $\gamma\in(0,1)$ and $c>0$ such that for any $n\ge1$ and any union $U\subset F_n$ such that $|U|\ge c\cdot |F_n|/n$ we can partition $U$ into $U'$ and $U''$ in such a way that
$$
\gamma \le \mu(U')/\mu(U'')\le 1/\gamma?
$$
If that's not necessarily true in this generality, what extra condition do we need?

For instance, if $X$ is a transitive subshift of finite type (or, more generally, a transitive sofic subshift) and $\mu$ is the unique measure of maximal entropy for $X$, then it is known that $\mu[x_1=i_1,\dots,x_n=i_n]\asymp \alpha^n$, where $\alpha=\exp(-h_\mu(\sigma))\in(0,1)$, so we can just split any $U$ into approximately equal parts (i.e., $||U'|-|U''||\le1$), and this'll do.

Of course, the Shannon-McMillan-Breiman theorem ensures that $\mu[x_1=i_1,\dots,x_n=i_n]\approx \alpha^n$ for *most* cylinders, but this is just too crude for my purposes, since the kind of subsets $U$ I'm dealing with have very small measure $\mu$.

**EDIT** added: a lower bound of the size of $U$.

**EDIT** #2 added: in view of Anthony's counterexample, let's assume $\mu$ to be a measure of maximal entropy for $X$. (Unique if it helps.)