The finite-unit subring conjecture for algebraic numbers

Let $R\le \overline{\mathbb{Q}}$ be a subring with finitely many units. An order conjecture. There exists an integer $d\in\mathbb{Z}_{<0}$ such that $R$ is an order in $\mathbb{Q}(\sqrt{d})$. The preceding proposition reduces the possibilities to this case or to a subring containing an element of the form $\frac{a+b\sqrt{d}}{c}$ with $c$ dividing $a^2-b^2d$; it is not clear whether that second case occurs, and the conjecture asserts that it does not.