The integer bounded-degree restricted-sumset conjecture

Let $\Delta\geqslant 1$ be an integer. Let $A,B\subseteq\mathbb{Z}$ satisfy $|B|\leqslant |A|$, and let $\mathcal{R}\subseteq A\times B$ be a binary relation. The maximum degree on $B$ is the largest number of elements of $A$ related to any single element of $B$. Define

$$A+_{\mathcal{R}}B=\{a+b:a\in A,\ b\in B,\ (a,b)\notin\mathcal{R}\}.$$

Bounded-degree restricted-sumset conjecture. If the maximum degree of $\mathcal{R}$ on $B$ is at most $\Delta$, then

$$|A+_{\mathcal{R}}B|\geqslant |A|+|B|-1-\left\lfloor\frac{5\Delta}{2}\right\rfloor.$$

The paper gives constructions attaining this bound, and presents the claim as a conjectural strengthening of its proved estimate; its general validity remains open.