The Ankeny–Artin–Chowla conjecture for real quadratic fields

Let $d\in\mathbb{N}_{\geq 2}$ be squarefree, let $K=\mathbb{Q}(\sqrt{d})$, and let $\mathcal{O}_K=\mathbb{Z}[\omega]$ be the ring of integers of $K$, where

$$\omega=\begin{cases}\sqrt{d}&\text{if }d\equiv 2,3\mod 4,\\frac{1+\sqrt{d}}{2}&\text{if }d\equiv 1\mod 4.\end{cases}$$

Let $\varepsilon=x+y\omega$ be the fundamental unit of $\mathcal{O}_K$ with $\varepsilon>1$, where $x,y\in\mathbb{N}_0$. Ankeny–Artin–Chowla conjecture. If $d\in\mathbb{P}$ and $d\equiv 1\mod 4$, then $d\nmid y$. This conjecture has been verified for all such primes through at least $1.5\cdot 10^{12}$, but the general assertion remains open.