Finding good modeling

Question context

Let $A,B$ be subset of abelian groups $G, H$ respectively. A map $\phi: A\rightarrow B$ is called a Freiman isomorphism if

We have the following modeling lemmas.

1. Let $A\subset \mathbb{F}_2^n$ and $|A+A|\leq K|A|$. Then $A$ is Freiman isomorphic to a subset of $\mathbb{F}_2^m$ where $|\mathbb{F}_2^m|\leq K^{O(1)}|A|$.

2. Let $A\subset \mathbb{Z}^d$ and $|A+A|\leq K|A|$. Then $A$ is Freiman isomorphic to a subset of a group $H$ where $|H|\leq K^{O(1)}|A|$.

Moreover, this is false for arbitrary groups instead of $\mathbb{Z}^d$ or $\mathbb{F}_2^n$.

Historical workshop record. Current open/solved status has not been independently reassessed.

The source supplies no status note for this question.