When is the canonical conductor minimal?

For a one dimensional analytically unramified Cohen-Macaulay local ring $R$, the blowup algebra of the canonical ideal is a module finite birational extension. The conductor of this extension always contains the conductor of $R$. We study the case when there is equality. This is the case where $R$ is far from being almost Gorenstein. We study this property within the landscape of numerical semigroup rings and local Arf rings.