The sumset obstruction conjecture for prime triple sumsets

Let $A,B,C$ be sets of positive integers, each with at least two elements, and suppose that $A+B+C$ consists entirely of primes. Write $(A+B+C)(x)$ for the number of elements of $A+B+C$ not exceeding $x$. Then

The sumset obstruction conjecture. If, for some $\kappa$, one has

$$(A+B+C)(x)\gg_\kappa\frac{x}{\log^\kappa x},$$

then at least one of the inequalities

$$|A+B|\leq\kappa,\qquad |A+C|\leq\kappa,\qquad |B+C|\leq\kappa$$

holds. The conjecture asserts that sufficiently large prime triple sumsets must therefore have a pairwise sumset of size at most $\kappa$. The paper presents this as a general conjecture after exhibiting Hardy–Littlewood constructions in the cases where one of these inequalities holds; no resolution is supplied in the provided text.