I'm reading about autoreducibility, which is the following notion:
A set $L$ is autoreducible if there is a polynomial-time oracle Turing machine $M$ that accepts $L$ using $L$ as an oracle, with the caveat that $M(x)$ may not query whether $x \in L$.
I encountered some results that are confusing me. In Using autoreducibility to separate complexity classes, Buhrman et al. showed that every polynomial-time Turing-complete set for EXPSPACE is autoreducible (Theorem 4.1). In Diagonalization, a survey by Fortnow, there is a theorem stating that if every Turing-complete set for EXPSPACE is autoreducible then $NL \neq NP$ (Theorem 3.1). So using both theorems together, we have $NP \neq NL$! I want to know where is my fault.