Hooley–Sarnak conjecture on fundamental units

Let $D\geq1$ be a fundamental discriminant, let $\mathbb{Q}(\sqrt D)$ be the associated real quadratic field, and let $\varepsilon_D$ denote its fundamental unit. Hooley–Sarnak conjecture. For every fixed $\varepsilon>0$, for almost every fundamental discriminant $D\geq1$,

$$\varepsilon_D>e^{D^{1/2-\varepsilon}}.$$

This is presented as a consequence of conjectures of Hooley and Sarnak and as an analogue for the elliptic-twist problem; the supplied text does not state that it has been proved.