Erdős–Selfridge odd covering problem

A covering system of the integers is a finite family of congruence classes $a_i \pmod{n_i}$ whose union is all of $\mathbb{Z}$. The moduli are odd, distinct, and greater than $1$ when every $n_i$ is odd, the moduli are pairwise distinct, and $n_i>1$ for every $i$.

Erdős–Selfridge odd covering problem. Does there exist a covering system of $\mathbb{Z}$ whose moduli are odd, distinct, and greater than $1$?

The problem is open in both directions and is Erdős's problem #7. An odd covering system would have consequences for the reducibility of trinomials, and the paper gives a kernel-checked exclusion showing that any such covering would have least common multiple exceeding $10000$.