Stevenhagen's conjecture on Eisenstein discriminants

Let $d$ be a positive square-free integer with $d\equiv 5 \bmod 8$, let $K=\mathbb{Q}(\sqrt{d})$ with ring of integers $\mathcal{O}_K=\mathbb{Z}[\frac{1+\sqrt{d}}{2}]$, and let $\varepsilon_d$ be its fundamental unit. Define

$$\mathcal{D}:=\{d\in\mathbb{Z}_{>0}\mid d\equiv 5\bmod 8,\ d\text{ is square-free}\}$$

and

$$\mathcal{E}:=\{d\in\mathcal{D}\mid \varepsilon_d\equiv 1\bmod 2\mathcal{O}_K\}.$$

Writing

$$\pi_{\mathcal{E}}(x):=\sum_{d\in\mathcal{E},\ d\leq x}1,$$

Stevenhagen's conjecture. As $x\to\infty$,

$$\pi_{\mathcal{E}}(x)\sim\frac{1}{3\pi^2}x.$$

The conjecture asserts that one third of the discriminants in $\mathcal{D}$ are Eisenstein discriminants, reflecting the three possible nonzero reductions of the fundamental unit modulo $2\mathcal{O}_K$.