The major-arcs conjecture for exceptional sets of additive problems

Let ${\mathcal A},{\mathcal B}\subseteq {\mathbb N}$. Write $\overline{{\mathcal A}+{\mathcal B}}$ for the exceptional set associated with the sumset ${\mathcal A}+{\mathcal B}$, let $|\mathcal B|_0^N$ denote the number of elements of ${\mathcal B}$ up to $N$, and let $\widehat {\Upsilon}(\overline{{\mathcal A}+{\mathcal B}},{\mathcal B};N)$ be the corresponding exceptional-set counting function. Major-arcs conjecture. One has

$$\widehat {\Upsilon}(\overline{{\mathcal A}+{\mathcal B}},{\mathcal B};N)\ll N^{\varepsilon-1}\left(\left|\overline{{\mathcal A}+{\mathcal B}}\right|_{2N}^{3N}|{\mathcal B}|_0^N\right)^2.$$

This estimate is suggested by a heuristic application of the Hardy–Littlewood method and would quantify the expected improvement in exceptional-set estimates obtained by adding ${\mathcal B}$. The source presents it as a speculation rather than a proved result.