OPEN

What is the minimal value of $\lvert 1-\sum_{n\in A}\frac{1}{n}\rvert$ as $A$ ranges over all subsets of $\{1,\ldots,N\}$ which contain no $S$ such that $\sum_{n\in S}\frac{1}{n}=1$? Is it
\[e^{-(c+o(1))N}\]
for some constant $c\in (0,1)$?

It is trivially at least $1/[1,\ldots,N]$.