It looks to me like it "disagrees" mostly by mis-stating what other interpretations predict about the (highly idealized) thought experiments it describes. For example, it says that if we had a "reversible machine intelligence", we could use it to entirely reverse a measurement after it was performed. But on a collapse interpretation (such as Copenhagen), such a "reversible" measurement, like any reversible process, is not a measurement at all; it doesn't collapse the wave function. So the interpretations don't actually differ on this prediction; they just differ on how the result (which they both agree on) is interpreted.
The FAQ does raise one issue which might amount to a "difference in prediction", namely the issue of whether linearity is exact or not. The MWI requires that it is; any collapse interpretation requires that it is not. However, even here there is a problem about what constitutes a "prediction". The MWI claims that the other "worlds" exist, but we can't communicate with them; we can only detect them via interference effects, which basically means by keeping processes reversible, as above, so a collapse interpretation will just say that no collapse has occurred. But that is effectively the same as the collapse interpretation's prediction that the other worlds don't exist after a collapse has occurred. In other words, these two putatively different "predictions" are in fact experimentally indistinguishable.
The only explicit experimental setup I can find on that webpage is in the last couple of paragraphs of Q.36, where it describes a "reversible machine intelligence" which can perform measurements and then reverse the entire measurement process. The issue is that this reversal requires having fine control over the joint quantum state of the system, measurement apparatus and all record of the measurement outcome. However, the very fact that this whole setup can be described as being in one quantum state or another is sufficient for a Copenhagen advocate to deduce that no collapse would have occurred.
Now, different flavours of Copenhagen will differ as to why collapse has not occured. Bohr would probably have argued that a measurement entails an interaction between a classical system and a quantum one, whereas this has not happened in the above. A more sophisticated approach like QBism would say that quantum theory is simply an agent's calculus of experience, and that if the entire system has a quantum representation then this means the agent is still external to the system and has not yet interacted with it. Either way, Copenhagen makes the same predictions as MWI in this experiment, as it does in all experiments that admit a purely quantum description.
It looks to me like it "disagrees" mostly by mis-stating what other interpretations predict about the (highly idealized) thought experiments it describes. For example, it says that if we had a "reversible machine intelligence", we could use it to entirely reverse a measurement after it was performed. But on a collapse interpretation (such as Copenhagen), such a "reversible" measurement, like any reversible process, is not a measurement at all; it doesn't collapse the wave function. So the interpretations don't actually differ on this prediction; they just differ on how the result (which they both agree on) is interpreted.
The FAQ does raise one issue which might amount to a "difference in prediction", namely the issue of whether linearity is exact or not. The MWI requires that it is; any collapse interpretation requires that it is not. However, even here there is a problem about what constitutes a "prediction". The MWI claims that the other "worlds" exist, but we can't communicate with them; we can only detect them via interference effects, which basically means by keeping processes reversible, as above, so a collapse interpretation will just say that no collapse has occurred. But that is effectively the same as the collapse interpretation's prediction that the other worlds don't exist after a collapse has occurred. In other words, these two putatively different "predictions" are in fact experimentally indistinguishable.