First, Cain's website doesn't say that. Second, is that even Constitutional? Currently the Constitution requires super majorities only for vetoes and for amendments. For everything else it requires only a simple majority. I don't think that Congress can just pass a law saying 2/3rds approval is required for some bills and not others.
Cain has said that in many interviews.
But that doesn't mean Cain's plan is any more likely to have the rates raised than the current system. In fact, it is less likely because:
(1) Because it's not a "progressive" tax rate. A rate increase affects everyone. No more class warfare calls for raising tax rates.
(2) It's transparent. The instant tax rates go up, everyone knows about it.
The biggest problem I forsee is the temptation of congress to slowly reintroduce deductions for various things. But that's the major problem with the current system (which is so big and complicated nobody can actually say how many pages are even in the tax code).