Thank you! I focused on the problem recently and used lingo to give an integer linear programming when I consider the 3-partition of 𝑆7 with 𝑇7 for 7-length sequences. However, the program ran for a long time without any results. Can you give the situation or suggestion at 7 so that I can try to give a general conclusion?

And the order on the index set should not be well-ordered,or it is obvious as the countable one.

It is an open problem. You can add some conditions