Structural conjecture for sumsets bounded by a fixed multiple of the set size

Let $\alpha$ be a real number satisfying

$$3<\alpha<3+\frac{1}{3},$$

and let $A\subseteq\mathbb N$ be finite. Write $2A=A+A=\{a+a':a,a'\in A\}$ and set $b=|2A|-3|A|+3$. An arithmetic progression is a finite set of equally spaced integers, and a bi-progression is the structured object used in the source's additive-combinatorial results.

Structural conjecture. There exists a $K\in\mathbb N$ such that for every finite set $A\subseteq\mathbb N$ with $|A|>K$, if

$$3|A|-3\leqslant |2A|=3|A|-3+b\leqslant\alpha|A|,$$

then $A$ is either a subset of an arithmetic progression of length at most $2|A|-1+2b$ or a subset of a bi-progression of length at most $|A|+b$.

The source presents this as a weaker version of its main conjecture, intended to be easier to prove and useful for improving a stated corollary. Its status evidence says that the paper's main theorem confirms only a weak version, so the conjecture remained unresolved there.