The only way Thr Rock keeps the name, and the rights therein, is if it is determined that it is a self-made, that is, he was the reason the name sticks in people's minds. You associate 'The Body' with Jesse Ventura, and 'Hitman' with Bret Hart (I remember the time when the announcer first announced Bret's adopted nickname. It was met with derision, as many people knew about Thomas Hearns). Also Hulk Hogan and derivative names with Terry Bollea. Peter will want to have a say about Dwayne Johnson being known excusively as 'The Rock'.
I'm gonna need evidence of this because the only PPV match I remember them having was Armageddon, which Jeffrey won clean. I don't doubt you, I just don't remember HHH beating Jeffrey in 2 to 3 straight pay per view matches.