Chabert–Halberstadt's Pólya group conjecture for S4-fields

Let $K$ be an $S_4$-field. The Chabert–Halberstadt conjecture.

$$\operatorname{Po}(K)=\operatorname{Cl}(K).$$

Here $\operatorname{Po}(K)$ is the Pólya group of $K$, and $\operatorname{Cl}(K)$ is its ideal class group. The conjecture asks whether every ideal class of an $S_4$-field is represented by a Pólya ideal; it is attributed to Chabert and Halberstadt.