The conjecture that the unit-group ring is trivial for non-big non-imaginary-quadratic fields

Let $K$ be a non-big field, meaning an algebraic extension of $\mathbb{Q}$ that is not big, and let $R_K$ denote the ring constructed from the unit group of $K$. Triviality conjecture. If $K$ is not an imaginary quadratic field, then

$$R_K=\mathbb{Z}.$$

Earlier results in the source show this in several cases, including degree-$4$ extensions and non-big fields without imaginary quadratic subfields or that are CM-fields. The conjecture remains open in the general non-big, non-imaginary-quadratic case.