Non-square Stark unit conjecture

Let $K^{\mathfrak p\mathfrak j}$ be the ray class field appearing in Theorem~, let $\epsilon_\sigma$ be its Stark unit, and let $\xi_\ell$ and $x_0$ be the associated quantities in that theorem. Non-square Stark unit conjecture. The Stark unit $\epsilon_\sigma$ is not a square. Consequently, the field extension in the theorem is equality:

$$K^{\mathfrak p\mathfrak j}(\sqrt{\epsilon_\sigma})=K^{\mathfrak p\mathfrak j}(\xi_\ell),$$

and

$$\sqrt{\epsilon_\sigma}\,\xi_\ell=\sqrt{x_0\epsilon_\sigma}\in K^{\mathfrak p\mathfrak j}.$$

The conjecture is motivated by computations in the listed prime dimensions and is stated as an arithmetic condition needed for the three main conjectures in the paper to have their concise form.