I have a question regarding non-cuspidal Hilbert modular forms. If one starts with a non-parallel weight for example, it is easy to prove that there are no Eisenstein series of any level, or as is generally stated, all forms are cuspidal. My question is what happens with mod p Hilbert modular forms? Are there (non-zero) non-cuspidal mod p Hilbert modular forms of non-parallel weight? (say at least when one or all the weights are greater than 1).

For classical modular forms, if the weight is greater than 1, the mod p modular forms are exactly the reduction of global modular forms, so the naive answer would be that there are none, but I am not too familiar with mod p Hilbert modular forms...