Erdős Problem #8 — Covering-system moduli inside one class of any finite partition

A family of residue classes $a_i \pmod{n_i}$ with $1 < n_1 < \ldots < n_r$ is called a system of covering congruences if every integer belongs to at least one of the residue classes, i.e., every integer satisfies at least one of the congruences $x \equiv a_i \pmod{n_i}$. [...] Is it true that if the positive integers are partitioned into finitely many classes then at least one of the classes contains the moduli of a covering system? Perhaps if a subset $X \subseteq \mathbf{Z}^+$ has positive upper density then $X$ already must contain the moduli of a covering system.