Must the surviving set of an arbitrary congruence sieve have a logarithmic density? (Erdős #25)

Let $1\le n_1<n_2<\cdots$ be an arbitrary strictly increasing sequence of moduli, and to each $n_i$ associate a residue class $a_i\pmod{n_i}$. Let $A$ be the set of integers $n$ such that for every $i$ either $n<n_i$ or $n\not\equiv a_i\pmod{n_i}$ — that is, $A$ consists of the integers that survive all of the congruence conditions that apply to them (a condition $a_i\pmod{n_i}$ only removes $n$ when $n\ge n_i$). Must the logarithmic density of $A$, namely $$\lim_{x\to\infty}\frac{1}{\log x}\sum_{\substack{n\in A\\ n\le x}}\frac1n,$$ exist for every such choice of moduli and residues?

Acceptance. FULLY RESOLVES: EITHER a complete proof that for every sequence of moduli $n_i$ and residues $a_i$ the logarithmic density of $A$ exists (a Lean/Coq-checkable proof preferred, otherwise a full written proof) — OR an explicit counterexample: a sequence $(n_i,a_i)$ together with a proof that $\frac{1}{\log x}\sum_{n\in A,\,n\le x}1/n$ has two distinct subsequential limits, so the logarithmic density does not exist. ADVANCES: prove that the logarithmic density exists under a stated structural restriction on the moduli (for example bounded moduli, or a growth/multiplicity condition), with a complete proof; OR establish the analogous existence statement for a natural weaker or stronger notion of density; OR give an explicit family of finite truncations whose survivor logarithmic-density partial sums are proved to oscillate by a quantified amount, delivered with the generating construction. Deliver the proof file, or the counterexample sequence plus the oscillation proof.