Also, in the proof of Lemma 5.3., bitwise uniform randomness is shown (already questionably), which is not sufficient for having a stationary distribution overall. Take for example X such that it is (0,0,...,0) or (1,1,...1) with probability 0.5 each. This is bitwise uniformly random, but is clearly not uniform on the whole cube.
Also, in the proof of Lemma 5.3., bitwise uniform randomness is shown (already questionably), which is not sufficient for having a stationary distribution overall. Take for example X such that it is (0,0,...,0) or (1,1,...1) with probability 0.5 each. This is bitwise uniformly random, but is clearly not uniform on the whole cube.