Conjecture on recurrence divisibility for algebraic integers

Let $\alpha \in \bar{\Z}$, let $K=\Q(\alpha)$, and let $d_k(\alpha)$ denote the largest positive integer such that

$$\alpha^k-1\in d_k(\alpha)\mathcal O_K.$$

Assume that either (a) $[\Q(\alpha^r):\Q]\geq 3$ for every $r\geq 1$, or (b) $[\Q(\alpha^r):\Q]\geq 2$ for every $r\geq 1$ and $N_K(\alpha)\ne\pm1$. Conjecture 9. Under these assumptions, the set

$$\{k\geq 1\mid d_k(\alpha)=d_1(\alpha)\}$$

is infinite. The conjecture predicts infinitely many indices at which the divisibility sequence attached to $\alpha$ returns to its initial value, outside the exceptional low-degree or quadratic-unit cases discussed in the paper.