Finite Birational extension with stable conductor

Let $S$ be a module finite birational extension of a $1$-dimensional local Cohen--Macaulay ring $R$. When is the conductor of $S$ in $R$ a stable ideal? If $R$ is also generically Gorenstein, then we show that the conductor of $S$ in $R$ is a stable ideal, and $S$ is a reflexive $R$-module if and only if $\Omega \operatorname{CM}(S)=\operatorname{CM}(S)\cap \Omega \operatorname{CM}(R)$.