The Desolation Row CP is not considered a permanent CP.
The only CP that count as permanent will use the word "permanent" on their card. A fight they'll never forget is a fine example. To elaborate further, Desolation Row actually has a trait that is constantly checking bounty and will give (or taketh away) when the requirements are met (or lost). While some dudes could hold onto it all game, it isn't considered permanent for card effects.