Intersective sets contain a difference of two primes

Let $\mathbb{P}$ denote the set of primes, and let $\mathbb{P}-\mathbb{P}=\{p-q:p,q\in\mathbb{P}\}$. A set $R\subset\mathbb{N}$ is intersective if it meets $S-S$ for every subset $S\subset\mathbb{N}$ of positive upper density.

Intersective prime-difference conjecture. For every intersective set $R$,

$$R\cap(\mathbb{P}-\mathbb{P})\neq\varnothing.$$

The paper explains that this would follow for every intersective set from the stronger expected inclusion $\mathbb{P}-\mathbb{P}\supset 2\mathbb{Z}$, but seeks the stated conclusion unconditionally. Its status is open.