Non-Parry powers conjecture for roots of $x^d-x-1$

For each integer $d\geq4$, let $4\beta_d4$ be the Perron root of

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

A Parry number is a real number greater than $1$ whose greedy expansion of $1$ is eventually periodic; a non-Parry number is a number that is not Parry.

Non-Parry powers conjecture. For every $d\geq4$ and every integer $k\geq1$, the power $4\beta_d^k4$ is non-Parry.

The conjecture is supported by computations, and the source remarks that it has been verified up to $d=100$. A proof is not given.