Hooley's conjecture on small fundamental units in real quadratic fields

Let $d$ range over nonsquare positive integers, and for fixed $\alpha>0$ define

S_\alpha(X)=\\#\left\\{d\leq X:\epsilon_d\leq d^{1/2+\alpha}\right\\}.

Here $\epsilon_d$ is the fundamental solution associated with the Pell equation for $d$.

Hooley's conjecture. For every $\alpha>0$, there exists a constant $b(\alpha)>0$ such that

$$S_\alpha(X)\sim b(\alpha)X^{1/2}(\log X)^2$$

as $X\to\infty$.

This conjecture seeks the asymptotic number of real quadratic fields whose fundamental unit is only slightly larger than its general lower bound. It is connected with the study of class numbers through the real quadratic class number formula; the source does not state a resolution.