Rationality of ξ₂ (the c=2 prime‑representing constant)

Determine whether ξ₂ is rational or irrational, where ξ₂ denotes the smallest real number ξ>1 such that ⌊ξ^{2^k}⌋ is a prime number for every positive integer k.

Background

For each integer c≥2, the paper defines ξ_c as the smallest real number ξ>1 such that ⌊ξ^{c^k}⌋ is prime for all k∈ℕ. The author proves that ξ_c is transcendental for every c≥4 and that ξ_3 is either transcendental or has a specific Pisot-number structure, which in particular implies ξ_3 is irrational.

The case c=2 is explicitly left unresolved, and the authors pose the foundational question of whether ξ₂, the minimal element of 𝒲(2), is rational or irrational.