Sorry but this story is bullshit. Jay-Z had already released Blueprint 2 by the time Pump It Up was recorded. Jay's freestyle over the Pump It Up beat was for his S. Carter Mixtape, the official prequel to The Black Album. Why didn't the story mention that? Beucase it's false.
Jay may have had problems with Clue, I'm not sure... but the fact is that Jay passed on the beat, Joey's camp requested Jay to be on the remix, Jay demanded a ridiculous amount (probably to spite someone... Clue?) and Joey said fuck it. When Jay's mixtape dropped, Joey definitely had to feel some type of way about being overcharged just for Jay to turn around and give a Pump It Up verse away for free, so Joe retailiated.
And if you listen to Jay's verse, some of his lines in there sound like they're directed at Budden. I wouldn't put it past Jay either since he's the king of subliminals.
