Conjecture on Siegel invariants generating fixed fields in ray class fields

Let $K$ be a CM-field with reflex field $K^*$, let $\mathfrak{f}$ be a modulus of $K$, and let $\mathfrak{f}^*$ be the corresponding modulus of $K^*$. Write

$$\widetilde{\varphi}:\mathrm{Cl}(\mathfrak{f})\longrightarrow\mathrm{Cl}(\mathfrak{f}^*)$$

for the natural homomorphism induced by the map $\varphi$ defined by the reflex norm, and let $C$ be the CM point used to define the Siegel invariant $\Theta_\mathfrak{f}(C)$. Siegel-invariant generator conjecture. The Siegel invariant $\Theta_\mathfrak{f}(C)$ is a primitive generator of the fixed field of $\ker(\widetilde{\varphi})$ in the ray class field $K_\mathfrak{f}$ of $K$.

This conjecture concerns the field generated by a Siegel invariant under the assumption referred to in the source. The displayed claim is formulated in terms of the ray class groups of $K$ and its reflex field; its resolution is not specified in the supplied text.