A cyclic-generator conjecture for principal $\ell$-adic units

Let $k=\mathbb Q(\sqrt{-d})$ be an imaginary quadratic field, let $K/\mathbb Q$ be Galois with group $G$ and containing $k$, put $H=G(K/k)$, and let $\tau\in G$ be complex conjugation, so that $\tau^2=1$ and $\tau(\sqrt{-d})\neq\sqrt{-d}$. Let $U_{S,1}(K)$ be the group of principal $\ell$-adic units used in the paper.

Cyclic-generator conjecture. There exists $\omega\in U_{S,1}(K)$ such that $\tau(\omega)=\omega$ and the set

$$\{h(\omega):h\in H\}$$

generates a submodule of finite index in $U_{S,1}(K)$.

The statement concerns the structure of principal $\ell$-adic units in Galois extensions containing an imaginary quadratic field. The supplied text gives no proof or resolution; the notation and the corrupted conjunction in the source should be checked.