Conjecture on squares in binary recurrence sequences

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 integers $x_k,y_k$ by

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

for $k\in\mathbb{Z}$, and let $\operatorname{sf}(n)$ denote the unique squarefree integer such that $n/\operatorname{sf}(n)$ is a square. Squares-in-one-sequence conjecture. There are at most four distinct integer squares among the $y_k$. If $\operatorname{sf}(|N_{\alpha}|)=2^{\ell}p^{m}$, where $\ell,m\in\{0,1\}$, $\ell+m\geq1$, and $p$ is an odd prime, then there are at most three distinct integer squares among the $y_k$. Furthermore, if $|N_{\alpha}|$ is a perfect square, then there are at most two distinct integer squares among the $y_k$. These bounds concern the number of square values occurring in the associated non-degenerate binary recurrence sequence; the conjecture is presented as part of the paper's study of quartic Diophantine equations and remains unresolved in the supplied source.