What if an employer cannot find the signed non-compete document?

I was terminated on September 18, 2015. I was not given a copy of my non-compete document with signature, which I don't remember signing. I requested, through an email chain, a copy of the signed document on 10/8, 10/15, and was given a response on 10/19 that a copy would be provided that day. I received nothing, so asked again today, 10/22, and received an email indicating that the President "based on her recollection" believes I signed the document. She stated that she is unable to locate the signed copy due to having files in multiple locations from moving offices and renovations, but she is confident that it will eventually be located. She will send it, but in the interim and based on her recollection I am to abide by the non-compete. Also, she stated that if, contrary to her recollection, and I did not sign the agreement, then the obligations contained therein would not govern. How long is she able to hold this over my head without providing the signed document to me? I am restricted within the industry from talking, networking, or pursuing anything in my field.