A small-doubling structural conjecture for finite sets of integers

Let $A$ be a finite set of integers, and write $A-A=\{a-a':a,a'\in A\}$. Small-doubling structural conjecture. There exists a natural number $K$ such that, whenever $|A|\ge K$ and

$$|A-A|=3|A|-3+r$$

for an integer $r$ satisfying $0<r<\tfrac13|A|-2$, one of the following holds: (i) $A$ is contained in an arithmetic progression of length $2|A|-1+2r$; or (ii) $A\subseteq P_1\cup P_2$ for arithmetic progressions $P_1,P_2$ with common step and $|P_1|+|P_2|\le |A|+r$. The source proposes this conjecture as a replacement for Jin's inverse theorem in order to obtain a reasonable value of a structural threshold. Its resolution is not stated, so it remains open.