When you say open I am not sure what that means. Likewise I am not sure what is your perspective, that of trustee or beney.
A trustee has authority over a trust's adminsitration and whether or not it is active or deemed complete. Sometimes trusts get docketed for one type of court supervision or another and then the court can have some say over that question. Not sure what's happening in your situation. But the trustee needs to hire a lawyer to give advice about fiduciary duty in this situation, and what the risks are for the trustee and also the beneficiary. That is a question that is not only a matter of state law, the law which may govern the trust, but also the potential venue where any collection might be initiated. Thus it is a complicated fact intensive question that will require specific advice based on the nature of the debts and property desired to be abandoned as well as a review of the trust instrument and laws related to all that.
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I suggest you obtain an opinion from a local, experienced estate and trust attorney that has reviewed the Trust and related documents regarding the property. It is very difficult to provide an opinion with the limited facts you have described.
Answers to questions on this site are not intended to be specific legal advice nor create an attorney-client relationship. Hiring an attorney is a very important process which requires a high degree of diligence as well as entering into an agreement regarding the services to be provided and the fees to be charged.