Composite values of floor(beta^{3^n}) for Pisot numbers of degree 3

Prove or disprove that for any Pisot number β of degree 3, the integers floor(β^{3^n}) are composite for infinitely many integers n.

Background

This problem is motivated by Mills’ constant ξ. If ξ were algebraic, then ξ^{3^m} would be a degree-3 Pisot number β for some m≥1, and the sequence floor(β^{3^n}) would be prime for all n ≥ 1 by Mills’ construction. An affirmative resolution of the problem would imply ξ is transcendental.

The paper surveys prior results about the compositeness of floor(α^n) for algebraic α. For Pisot and Salem numbers α, it is known that floor(α^n) is composite infinitely often; however, the case with exponentially growing exponents like 3^n is significantly harder and akin to the difficulties surrounding Fermat numbers. The authors extend results to iterated linear recurrence sequences but note that this particular case remains beyond their methods.