Green's Open Problem 52

Reference: Green's Open Problems

theorem Green52.green_52 : sorry ↔ ∃ (c : ℕ → ℕ), ∀ (n K : ℕ) (A : Set (𝔽₂ n)) (S : Finset (𝔽₂ n)), S.card = K → A + ↑S = Set.univ → ∃ (V : AffineSubspace (ZMod 2) (𝔽₂ n)), ↑V ⊆ A + A ∧ n ≤ Module.finrank (ZMod 2) ↥V.direction + c K

Suppose that $A \subset \mathbb{F}_2^n$ is a set with an additive complement of size $K$. Does $2A$ contain a coset of codimension $O_K(1)$?

theorem Green52.green_52_log : sorry ↔ ∃ (C : ℝ) (D : ℝ), ∀ (n K : ℕ) (A : Set (𝔽₂ n)) (S : Finset (𝔽₂ n)), 0 < K → S.card = K → A + ↑S = Set.univ → ∃ (V : AffineSubspace (ZMod 2) (𝔽₂ n)), ↑V ⊆ A + A ∧ ↑n ≤ ↑(Module.finrank (ZMod 2) ↥V.direction) + C * Real.log ↑K + D

Could $2A$ even contain a coset of codimension $O(\log K)$?