Let $X$ be a normal connected complex analytic space, $x\in X$ a point, $f$ a nonzero holomorphic function vanishing at $x$. Denote by $U\subseteq X$ the locus where $f$ is nonzero. Suppose that $\pi:U'\to U$ is a finite degree covering space. This gives $U'$ the structure of an analytic space. Let $A$ be the ring of pairs $(V, h)$ where $V$ is a neighborhood of $x$ in $X$ and $f$ is a **bounded** holomorphic function on $\pi^{-1}(V\cap U)$, where for $V'\subseteq V$ we identify $(V, h)$ with $(V', h|_{V'})$. This is a subring of the stalk of $j_* \pi_* \mathcal{O}_{U'}$ at $x$, where $j:U\to X$ is the inclusion. It contains $\mathcal{O}_{X, x}$ (the ring of germs of holomorphic functions at $x$).

Question.Is $A$ a finitely generated $\mathcal{O}_{X, x}$-module?

In other words, can we "normalize $X$ inside $U'$", that is, extend $U'\to U$ to a finite map $X'\to X$?