[jimmydef]

Baozi is unlikely to have come from the Ottomans via the Mongols.

The Song dynasty chinese were already eating baozi and writing about it in their books and documents in the early 11th century, before the Ottomans were established or Genghis Khan was born.

And Baozi is just another name for Mantou in ancient China which is a name for the food which is even older stretching back to the Han dynasty over two millenia ago.

[contingencies]

OK, maybe you are right but {{citation-needed}}. It seems Wikipedia asserts Song but has no references. As they are sequential the big picture is still similar. Song was very multicultural/pluralistic, even if thry appeared in the Song this virtually supports an import hypothesis.