Can you be placed in a university which you did not obtain LOA when you already have LOA from other university?

Hello!

I have obtained recently an LOA from a private university, but was rejected from a national one. Since I applied to only these schools, I couldn't apply to my third choice.

So my question is: can you place an another national university as a first choice in a placement preference form, and put a private one as a second choice even though I obtained an LOA from them?

I have heard that MEXT will try to place you in universities those that you indicated, but will prioritize national over private ones. How would it work in this case? Have anyone heard about such cases?

Thank you for answers