Generalized conjecture on squares in the associated recurrence sequence

Let $a$, $b$, and $d$ be positive integers with $d$ nonsquare, 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 $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ with $t,u$ positive integers. Define $x_k,y_k$ by replacing $\varepsilon^{2k}$ in

$$x_k+y_k\sqrt{d}=\alpha\varepsilon^{2k}$$

with $\varepsilon^k$, so that $y_k$ is the coefficient of $\sqrt d$ in $\alpha\varepsilon^k$. Generalized 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 generalizes the first bound to the sequence obtained using all powers of the unit rather than only even powers, and remains unresolved in the supplied source.