You need to get a free consultation with an attorney. There are too many factors involved in a custody/child support decision to be able to evaluate your question properly. Get in with an attorney, tell them whats going on, and they can advise as to how seriously you need to take any threats he may be making.
Clearly, to file for custody, there must be some underlying basis that he is alleging. On the surface, his case does not look to be compelling, but in fairness, I cannot advise you to simply ignore his potential claim. Go see a local attorney, consult with him or her, and do not be shy about letting counsel be aware of any potential issues which might be raised in the litigation. Never underestimate an opponent. Obtain good counsel, and that's how you win. Be prepared. Good luck.
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