Chabert–Halberstadt's pre-Pólya group conjecture for D4-fields of class number two

Let $K$ be a $D_4$-field, where $D_4$ denotes the dihedral group of order $8$, and let $h_K$ denote the class number of $K$. The pre-Pólya group $\operatorname{Po}(K)_{nr}$ is the subgroup of $\operatorname{Cl}(K)$ generated by the classes of Ostrowski ideals above unramified primes.

Chabert–Halberstadt's conjecture. If $h_K=2$, then

$$\operatorname{Po}(K)_{nr}=0.$$

This is a special conjecture about the pre-Pólya group of quartic $D_4$-fields with class number two. It was proposed by Chabert and Halberstadt, and the source gives no resolution.