Existence of odd distinct covering systems

Determine whether there exists a distinct covering system in which every modulus is odd.

Background

A distinct covering system is a finite collection of congruences with pairwise distinct moduli that covers every integer. The paper identifies the existence of a distinct covering system whose moduli are all odd as a longstanding question attributed to Erdős and Selfridge. Although several necessary conditions are known—for example, some modulus must be divisible by 3, and stronger divisibility restrictions involving 9 and 5 have been established—the general existence question remains unresolved.