Open. First question: only a weak version is known, a nonzero signed sum of absolute value at most $2^{-n(\log\log\log n)^{1+o(1)}/\log n}$ (site commentary crediting Kovač and van Doorn; comment arguments resting on the refereed count of distinct reciprocal subset sums of Problem 320), far from $c/2^n$, and a heuristic in the thread suggests the weak bound may be the truth. Second question: the strict inequality fails at $n=4$ (the monograph's example); no proof for all large $n$ exists, and even its special case for sums of the form $1-\sum_{n\in A}1/n$ is described as nontrivial in the thread of Problem 311. No source beyond the monograph and the site's commentary was found in the search whose scope the Current assessment records; this is a bounded negative finding.