Coprime sets in $[1,n)$: is $\sum_{a\in A}1/(n-a)\leq\sum_{p<n}1/p+O(1)$? (Erdős #1210)

Let $A\subseteq[1,n)$ be a set of integers that is pairwise coprime, meaning $\gcd(a,b)=1$ for all distinct $a,b\in A$. Is it true that $$\sum_{a\in A}\frac{1}{n-a}\leq\sum_{p<n}\frac{1}{p}+O(1),$$ where the right-hand sum runs over primes $p<n$ and the $O(1)$ is an absolute constant, independent of both $n$ and the set $A$?

Acceptance. FULLY RESOLVES: a proof (machine-checkable Lean/Coq preferred, else a complete written proof) that there is an absolute constant $C$ with $\sum_{a\in A}1/(n-a)\leq\sum_{p<n}1/p+C$ for every $n$ and every pairwise-coprime $A\subseteq[1,n)$; OR a disproof exhibiting, for a sequence $n\to\infty$, pairwise-coprime sets $A_n\subseteq[1,n)$ with $\sum_{a\in A_n}1/(n-a)-\sum_{p<n}1/p\to\infty$, together with a certificate that each $A_n$ is pairwise coprime and a proof the difference is unbounded. ADVANCES: (a) prove the inequality with an explicit sub-constant slack in place of $O(1)$, e.g. $\sum_{a\in A}1/(n-a)\leq\sum_{p<n}1/p+o(\log\log n)$ or $\leq(1+o(1))\sum_{p<n}1/p$, with proof; (b) establish the [Er77c] interval version for the primes in $(n,m]$, with proof; (c) exhibit, via extremal search, pairwise-coprime sets attaining the largest known value of $\sum_{a\in A}1/(n-a)-\sum_{p<n}1/p$, supplying the search code and the certified difference as evidence for the true constant. Deliver the proof, the counterexample family plus coprimality certificate, or the extremal-search code plus attained differences.