Which sequences b_n admit a primitive sequence a_n growing no faster than b_n? (Erdős #892)

Call a sequence $a_1<a_2<\cdots$ of positive integers primitive if no term divides another. (i) Is there a necessary and sufficient condition on an increasing sequence $b_1<b_2<\cdots$ that guarantees the existence of a primitive sequence $a_1<a_2<\cdots$ with $a_n\ll b_n$ for all $n$? In particular, is such a primitive sequence always guaranteed when the $b_i$ admit no non-trivial solution of $\gcd(b_i,b_j)=b_k$? (ii) Analogously, find necessary and sufficient conditions on a sequence $n_1<n_2<\cdots$ ensuring the existence of a primitive set $A$ with $$\lvert A\cap[1,2^{n_i}]\rvert\gg 2^{n_i}\quad\text{for every }i.$$

Acceptance. FULLY RESOLVES: (a) settle the concrete sufficiency conjecture — a full proof that every increasing $\{b_n\}$ with no non-trivial solution of $\gcd(b_i,b_j)=b_k$ admits a primitive sequence $a_n\ll b_n$, or an explicit counterexample sequence together with a proof that no such primitive sequence exists; OR (b) a proved genuine necessary-and-sufficient characterisation for either part (i) or part (ii). ADVANCES: a new necessary condition, or a new sufficient condition, strictly sharper than the two known necessary conditions stated in background (with proof); or a proof of the $\lvert A\cap[1,2^{n_i}]\rvert\gg 2^{n_i}$ characterisation for a natural, clearly specified class of sequences $\{n_i\}$. All artifacts must be machine- or referee-checkable (a proof, or a counterexample with a proof of its failure); a finite-prefix search may support a conjecture but does not resolve it. Deliver the proof, the characterisation with proof, or the counterexample with proof.