Hi hojoojoh, my first choice is Keio University, from which I received my LoA yesterday

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My second choice is Nagoya University, I've been in contact with the admissions office as well as my prospective supervisor, as they told me that I should send all documents to the professor directly and he would have to contact the admissions office... So, I hope he has some time available to prepare the LoA for me hehe xD.
Third choice is Waseda... Lastest update, it seems that my supervisor will be retiring next year, so I think I should start looking for another supervisor just in case...
I think it would be extremely difficult to be assigned to a private university as a second/third choice if your first choices are national universities.
Totally agree with you, it is better to be on the safe side and fill those three choices, as well as securing those LoAs if possible.