Conjecture 20.76

by L. Pyber Reference: The Kourovka Notebook !

theorem Kourovka.«20.76».kourovka.«20.76» : sorry ↔ ∀ (p : ℕ), Nat.Prime p → ∀ (G : Type) (x : Group G), IsPGroup p G → Finite G → ∀ (k : ℕ), (∀ (H : Subgroup G), H.Normal ∧ IsMulCommutative ↥H → Nat.card ↥H ≤ p ^ k) → ∀ (H : Subgroup G), IsMulCommutative ↥H → Nat.card ↥H ≤ p ^ (2 * k)

Let $G$ be a finite $p$-group and assume that all abelian normal subgroups of $G$ have order at most $p^k$. Is it true that every abelian subgroup of $G$ has order at most $p^{2k}$?