Let $X$ be a random variable taking values in $\Sigma^n$ (for some large alphabet $\Sigma$), which has very high entropy - say, $H(X) \ge (n- \delta)\cdot\log|\Sigma|$ for an arbitrarily small constant $\delta$. Let $E \subseteq \rm{Supp}(X)$ be an event in the support of $X$ such that $\Pr[X \in E] \ge 1 - \varepsilon$, where $\varepsilon$ is an arbitrarily small constant.
We say that a pair $(i,\sigma)$ is a low probability coordinate of $E$ if $\Pr[X \in E | X_i = \sigma] \le \varepsilon$. We say that a string $x \in \Sigma^n$ contains a low probability coordinate of $E$ if $(i, x_i)$ is a low probability coordinate of $E$ for some $i$.
In general, some strings in $E$ may contain low probability coordinates of $E$. The question is can we always find a high probability event $E' \subseteq E$ such that no string in $E'$ contains a low probability coordinate of $E'$ (and not of $E$).
Thanks!