Ben Green's Open Problem 60

Reference: [Ben Green's Open Problem 60](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.8 Problem 60)

theorem Green60.green_60 : sorry ↔ ∃ c > 0, ∀ (A : Finset ℕ), (∀ a ∈ A, IsSquare a) → 2 ≤ A.card → ↑(A + A).card ≥ ↑A.card ^ (1 + c)

Is there an absolute constant $c > 0$ such that, whenever $A ⊆ \mathbb{N}$ is a set of squares with $|A| ≥ 2$, the sumset $A + A$ satisfies $|A + A| ≥ |A|^{1 + c}$?