Freiman's structural conjecture for sets with doubling near three times their size

Let $A$ be a finite set of natural numbers, write $|A|=k$, and let $2A=A+A=\{a+a':a,a'\in A\}$. An arithmetic progression is a finite set of equally spaced integers, and a bi-progression is the corresponding structured object used in Freiman's theorem.

Freiman's conjecture. There exists a natural number $K$ such that for any finite set of natural numbers $A$ with $|A|=k>K$ and

$$|2A|=3k-3+b,\qquad 0\leqslant b<\frac{1}{3}k-2,$$

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

This conjecture seeks a precise structural description of large finite sets whose sumset has size just above three times the size of the original set. The surrounding discussion records earlier partial results of Freiman and indicates that the conjecture was unresolved in the source.