The Conn that beat Bob Pastor in 1940 would beat the version of RJJ that showed up to fight Antonio Tarver in their second fight. In this scenario I give the edge of speed to RJJ in the early rounds but Conns stamina would prevail especially if its a 15 rounder.
Best for best, I think that Roy would get home with the decision more often than not. He'd be dealing with the "heavyweight" Conn though and that guy should not be underestimated as a puncher, nor as a guy who could put you in trouble and keep you there, even if he didn't knock you out. Conn could beat Roy. But that wouldn't be my pick.
Toss it really. Roy was an offensive juggernaut with reflexes and savvy to get by defensively, but Conn was as tricky and clever as they come, and he did put a dent in Louis.
There was a time Roy was pretty much untouchable. He was too fast, too unorthodox, and punched too hard for just about anyone. If Conn was a big puncher, he'd at least have something to equalize all that, but he was not big hitter, (although it's true he did rock Louis, but I think this says more about Joe's chin than it does Conn's power). I don't see Conn bringing anything that Roy doesn't do better. I go with Roy to win a decision.