3 problems

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.

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

Dynamical strong Lehmer conjecture. If $4\beta<\tau_{10}4$ is a Perron number, then $4\beta^n4$ is a Parry number for only finitely many integers $4n4$.