The weak AACM conjecture for fundamental units of real quadratic fields

Suppose $d>0$ is a square-free integer and write $d=pm$ for an odd prime $p$ and integer $m$. If $d\equiv 1\left(\bmod{\,4}\right)$, set $\delta=1$; otherwise let $\delta=2$. Let $\varepsilon=\frac{\delta}{2}\left(t+u\sqrt{d}\right)$ be the fundamental unit of $\mathbb{Q}(\sqrt{d})$. The weak AACM conjecture. There exists an integer $\kappa>1$, independent of $d$, such that $d^\kappa\nmid u$. The original AACM conjectures are known to have counterexamples, so this weaker formulation is proposed as a conjectural uniform restriction on the divisibility of the coefficient $u$ in fundamental units; its resolution is not supplied here.