In the absence of any information whatsoever, the idea that an analysis pipeline produces false positives at or below its nominal rate seems pretty reasonable.
But, they have some prior information.
Let's look at the E1 paradigm (2 sec on, 6 sec off). In the NeuroImage Paper (Figure 1A, 2A), the FWER on voxel tests is statistically indistinguishable from 5%. In other words, it's appropriately sized. They replicate this result in the rightmost panel of the PNAS paper, where it's also within the 95% CI around 5%.
Now, for the cluster inference, look at the left-most panel of PNAS Figure 1A. The E1 paradigm is the green bar. Using the defaults for FSL (left panel) and SPM (middle panel), the FWER is about 30% and 25%, respectively. That is not good.
I agree that the Block designs look awful in the NeuroImage paper, which makes it hard to say whether this phenomena makes it worse in the PNAS data. It's unfortunate that these numbers are going to be in the press release (70% is much sexier than 30%), but 6x the nominal rate is still bad.