"Allege" is a close one, but it related to something illegal or wrong. Is there a word just means "assert, but without or before proof."?
Alleged, as an adjective, means that something was said to have taken place, but it has not been proven. It is often used when reporting about a person or incident that occurred, but the person has not yet been tried and convicted of the crime or the incident has not been verified by authorities. Unfortunately it is frequently used incorrectly. In your first example sentence, alleged means "asserted to be true, often without or before proof." -- An answer about alleged on english.stackexchange