Lev's matching restricted-sumset conjecture

Let $p$ be a prime number, and let $A,B\subseteq\mathbb{F}_p$ with $|B|\leqslant |A|$. Let $\mathcal{R}\colon B\to A$ be an injective function, viewed as a matching between subsets of $A$ and $B$. For the restricted sumset

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

Lev's conjecture.

$$|A+_{\mathcal{R}}B|\geqslant \begin{cases} |A|+|B|-3&\text{if }|A|+|B|\leqslant p,\\ p-3&\text{if }|A|+|B|=p+1,\\ p-2&\text{if }|A|+|B|\geqslant p+2. \end{cases}$$

The conjecture concerns the minimum size of a sumset after deleting the pairs specified by a matching; the paper confirms it when $|A|+|B|\leqslant(1-\varepsilon)p$ for sufficiently large $p$, but the full statement remains open.