# Is every Turing complete set for EXPSPACE autoreducible?

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.