Ostmann's inverse Goldbach conjecture

Let $\mathscr P$ denote the set of all primes. For sets $S,T$ of positive integers, define $S\sim T$ to mean that their symmetric difference is finite. Ostmann's inverse Goldbach conjecture. There do not exist sets $\mathcal A,\mathcal B$ of positive integers, each with at least two elements, such that

$$\mathscr P\sim \mathcal A+\mathcal B.$$

This is the inverse form of the Goldbach problem: it asks whether the primes can be, up to finitely many exceptions, represented as the sumset of two nontrivial sets. The source presents it as an important motivation for the paired inverse large sieve conjecture; its resolution status is not specified in the supplied text.