The four-squares conjecture for unrestricted unit-power Pell sequences

Let $a$, $b$ and $d$ be positive integers such that $d$ is not a square, let $\alpha=a+b^{2}\sqrt{d}$ have norm

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

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

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

Unrestricted-power 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 $y_k'$. This is a proposed strengthening of the even-power sequence conjecture, obtained by allowing all integer powers of the unit; it likewise concerns square values in families of solutions to the generalized Pell equation $x^2-dy^2=N_{\alpha}$.