Weber's class number problem in the cyclotomic ℤ₂-extension of ℚ

Let h_n denote the class number of ℚ(2cos(27π/2 ⁿ⁺²)). Weber proved that h_n is odd for all n ≥ 1. We claim that if I is a prime number less than 10⁷, then for all n≥ 1, ℓ does not divide h_n.