The main step seems to be Lemma 2. The fact that the reductions are logspace is never actually used in the argument - so if this proof worked, it would also app

The main step seems to be Lemma 2. The fact that the reductions are logspace is never actually used in the argument - so if this proof worked, it would also apply to arbitrary reductions, and the same argument would show that NP is not contained in, say, EXP. The fact that there are no inequalities anywhere throughout the paper is the real tip-off that this is not a serious attempt.