There's a (fairly weak) tendency for remain to be used of states, conditions, whereas stay is more likely when referring to locations, attitudes.
Perhaps linked to that, stay tends to be used where the subject is in control of the attribute/situation continuing without change. Thus,...
1: I remained unmarried until I was 30.
2: I stayed unmarried until I was 30.
...where I think #1 is more likely if the speaker would have liked to marry earlier (but circumstances didn't work out as he might have hoped). And #2 is more likely if the speaker deliberately chose to enjoy his 20s as a footloose and free bachelor.
For both those reasons, it's better to say the author remains unknown in OP's context. To me, the author stays unknown suggests the author is taking active steps to avoid detection.
For what it's worth, here are a few more contexts where the two words aren't likely to be thought of as "interchangeable". But I wouldn't like to say how much they reflect underlying principles, as opposed to randomly-established idiomatic preference...
3: Stay back! (injunction not to come any closer)
4: Don't stay out late! (injunction to be back reasonably early)
5: I ate everything he gave me, but I remained hungry.
6: I begged him to reconsider, but he remained implacable.