Low-dimensional subset conjecture for sets with small doubling

Let $A \subseteq \mathbb{Z}$ be a finite set with $|A+A| \le K|A|$.

Low-dimensional subset conjecture. There is a subset $X \subseteq A$ such that

$$|X| \ge \frac{|A|}{K^{O(1)}}\quad\text{and}\quad \dim(X)=O(\log K).$$

This is presented as a model problem toward polynomial bounds in Freiman's theorem; the source gives no resolution, so the conjecture remains open.