Three-translate linear stability conjecture for finite subsets of the integers

Let $A$ and $B$ be finite subsets of $\mathbb{Z}$ of equal size $n$. For $b_1,b_2,b_3\in B$, write $A+\{b_1,b_2,b_3\}=\{a+b_i:a\in A,\,i\in\{1,2,3\}\}$. For finite sets $X,Y$, write $X\mathbin{\Delta}Y$ for their symmetric difference.

Three-translate linear stability conjecture. For every $\delta>0$ there exists $\varepsilon>0$ such that, if

$$|A+\{b_1,b_2,b_3\}|\leqslant (2+\varepsilon)n-1$$

for every $b_1,b_2,b_3\in B$, then there exist arithmetic progressions $P,Q$ in $\mathbb{Z}$ with the same common difference such that

$$B\subset Q\qquad\text{and}\qquad |A\mathbin{\Delta}P|,\ |B\mathbin{\Delta}Q|\leqslant\delta n.$$

This is the proposed three-translate analogue of the paper's inverse theorem for four translates. The source presents this weaker stability form as conjectural, while noting that a stronger version is also expected.