Player's intent is the duration
You are using the word "early" to justify the inclusion of 1 minute. You are wrong. As persistent rage also has a neat word added to it; the word "only".
Now, since Persistent Rage's description does not include a duration, no duration should be applied. Rules as written the only two things that end a rage are the unconscious state or you choosing to end it. The word early, although may imply a set duration, doesn't list the set duration.
Example, if I wanted to rage for a week straight, long resting and everything (which is allowable since long resting can be sleeping or light activity) then boom I get knocked unconscious, my rage would end earlier than my intent.
My point is, since no duration is set for Persistent Rage. Then the players intent is the duration.