Parry and non-Parry powers conjecture for roots of $x^d-x^{d-1}-1$

For each integer $d\geq2$, let $4\theta_d4$ be the relevant root of

$$x^d-x^{d-1}-1=0.$$

For $d\equiv5\pmod 6$, use the noncyclotomic factor as specified in the source. A Parry number is a real number greater than $1$ whose greedy expansion of $1$ is eventually periodic.

Parry-power conjecture. For every $d\geq6$, $4\theta_d4$ is Parry, and for every integer $k\geq2$, $4\theta_d^k4$ is non-Parry.

The claim is introduced after computational tests for the family and is presented as a conjecture. No proof or resolution is supplied.