gn.general topology – Does every open set contain a dense $F_{sigma}$ subset?

Let $U$ be a regular open set in a Tychonoff space $X$ (regular means that it is an interior of a closed set).

( In my specific situation $U$ is of the form $int f^{-1}(0)$, where $f$ is a continuous real-valued function on $X$, and $X$ is a Baire space (a sequence of dense open sets has a dense intersection), but I am not sure if it helps. )

Is there a sequence ${A_n}_{ninmathbb{N}}$ of closed (in $X$) subsets of $U$ such that $bigcup_{ninmathbb{N}} A_n$ is dense in $U$?

Of course, this is the case if $X$ is perfectly normal (which is equivalent to every open set being $F_{sigma}$), or separable, but I hope a less restrictive assumption will suffice, e.g. normality.