Erdős Problem #429

Is it true that, if \(A\subseteq \mathbb{N}\) is sparse enough and does not cover all residue classes modulo \(p\) for any prime \(p\), then there exists some \(n\in \mathbb{Z}\) such that \(n+a\) is prime for all \(a\in A\)? Weisenberg [We24] has shown the answer is no: \(A\) can be arbitrarily sparse and missing at least one residue class modulo every prime \(p\), and yet \(A+n\) is not contained in the primes for any \(n\in \mathbb{Z}\). (Weisenberg gives several constructions of such an \(A\).)