Can a sub-sum of reciprocals approach 1 from below within e^{-cK} once the mass exceeds K? (Erdős #312)

Does there exist a constant $c>0$ such that, for every $K>1$, whenever $A$ is a sufficiently large finite multiset of positive integers with $\sum_{n\in A}\frac{1}{n}>K$, there is a sub-multiset $S\subseteq A$ with $$1-e^{-cK}<\sum_{n\in S}\frac{1}{n}\le 1?$$ In other words, can one always approximate $1$ from below by a sub-sum of the reciprocals to within an error that is exponentially small in $K$, once the total reciprocal mass exceeds $K$?

Acceptance. FULLY RESOLVES: a complete written (or machine-checked) proof that such a $c>0$ exists — that a from-below gap $1-\sum_{n\in S}\frac1n<e^{-cK}$ is always achievable for sufficiently large $A$ with $\sum\frac1n>K$ — or a proof that no exponential gap is possible (that some sub-exponential rate — e.g. Korsky's $\exp(-c\sqrt{K\log K})$, or another intermediate function — is best possible). ADVANCES (each independently checkable): (a) improve the known achievable from-below gap strictly beyond Korsky's stretched-exponential $\exp(-c\sqrt{K\log K})$ (e.g. to $\exp(-cK/\log K)$, the full $\exp(-cK)$, or any explicitly smaller function), with proof; (b) exhibit an explicit family of multisets, with a verified subset-sum computation and a proof that it holds for all large members, forcing the from-below gap to remain $\ge g(K)$ for a stated $g$, thereby bounding how small $c$ can be. Deliver the proof, the improved-rate proof, or the explicit family + verification.