Let $(R,m)$ be a Noetherian local domain whose integral closure $S$ is too. Also assume that $S$ is module-finite over $R$.

Let $x\in m^k\setminus m^{k+1}$ and $u\in S^\times$ such that $ux \in R$. Is $ux$ necessarily in $m^k\setminus m^{k+1}$?

The case I care about is when $R$ is an affine semigroup ring over $\Bbb C$ with maximal (multigraded) homogeneous ideal $m$ (or rather the localization of this stuff at $m$).