Dynamical strong Lehmer conjecture for Parry powers

Let $4\tau_{10}\approx 1.1764$ be the Salem number defined by the minimal polynomial

$$L(x)=x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1.$$

A Perron number is a real algebraic integer greater than $1$ whose other algebraic conjugates have strictly smaller modulus, and a Parry number is a real number greater than $1$ whose greedy expansion of $1$ is eventually periodic.

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 $4n4$.

The claim is presented as a dynamical version of the strong Lehmer conjecture and is motivated by the preceding power results. Its resolution is not specified in the source.