The four-squares conjecture for unrestricted unit-power sequences

Let $a$, $b$ and $d$ be positive integers such that $d$ is not a square. Set

$$\alpha=a+b^{2}\sqrt{d},\qquad N_{\alpha}=a^{2}-b^{4}d,$$

and let $\varepsilon=(t+u\sqrt{d})/2$ be a unit in $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ with $t,u$ positive integers. Define sequences $(x_k')_{k\in\mathbb{Z}}$ and $(y_k')_{k\in\mathbb{Z}}$ by

$$x_k'+y_k'\sqrt{d}=\alpha\varepsilon^k.$$

The unrestricted four-squares conjecture. There are at most four distinct integer squares among the $y_k'$. If $|N_{\alpha}|$ is a prime power or a perfect square, then there are at most three distinct integer squares among the $y_k'$. The conjecture extends the even-power sequence problem to all powers of the unit; the special norm conditions predict a stronger bound, but no resolution is supplied in the source.