Is it true that for every $\epsilon>0$ there exists some $k$ such that the density of integers not satisfying any of the congruences $a_i\pmod{n_i}$ for $1\leq i\leq k$ is less than $\epsilon$?

OPEN

Let $n_1<n_2<\cdots$ be an infinite sequence with associated congruence classes $a_i\pmod{n_i}$ such that the set of integers not satisfying any of the congruences $a_i\pmod{n_i}$ has density $0$.

Is it true that for every $\epsilon>0$ there exists some $k$ such that the density of integers not satisfying any of the congruences $a_i\pmod{n_i}$ for $1\leq i\leq k$ is less than $\epsilon$?

The latter condition is clearly sufficient, the problem is if it's also necessary. The assumption implies $\sum \frac{1}{n_i}=\infty$.