The Weak Anti-Freiman conjecture

Let $A \subseteq \mathbb Z$ be a finite set. For every subset $A' \subseteq A$, suppose that

$$|A'+A'|\geq \delta |A'|^2 \quad\text{and}\quad |A'-A'|\geq \delta |A'|^2.$$

Here, a $B_2[g]$ set is a set in which every integer has at most $g$ representations as a sum of two elements, and a $B^\circ_2[g]$ set is a set in which every nonzero integer has at most $g$ representations as a difference of two elements.

Weak Anti-Freiman conjecture. Then $A$ contains either a $B_2[g]$ set or a $B^\circ_2[g]$ set of size at least $\delta'|A|$, where $g$ and $\delta'$ depend only on $\delta$.

The conjecture is presented as a weaker version of an anti-Freiman question, but the paper states immediately afterward that it is false and constructs counterexamples.