Let $(R, \mathfrak m)$ be a local Cohen-Macaulay domain which is analytically unramified (i.e. the $\mathfrak m$-adic completion of $R$ is reduced). Let $\bar R$ be the integral closure of $R$ in the fraction field of $R$ (so $\bar R$ is module finite over $R$).
Let $C_R:=\{r\in R: r\bar R \subseteq R\}$.
My question is: Can $C_R$ ever be contained in a parameter ideal ?
(An ideal $I$ is called a parameter ideal if $\sqrt I=\mathfrak m$ and $\mu(I)=\dim R$ ).