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 $n$ such that $n+a$ is prime for all $a\in A$?

1False ↔2  ∃ f,3    Filter.Tendsto f Filter.atTop Filter.atTop ∧4      ∀ (A : Set ℕ),5        A.Infinite →6          (∀ (N : ℕ), (A ∩ Set.Icc 1 N).ncard ≤ f N) →7            (∀ (p : ℕ), Nat.Prime p → ∃ b, ∀ a ∈ A, ↑a ≠ b) → ∃ n, ∀ a ∈ A, Nat.Prime (n + a)