$P \neq NP$ and determinism Suppose $P \neq NP$. Does it imply that there exists some superpolynomial time bound, such that any $NP$-complete problem, like SAT, can be used to simulate an arbitrary deterministc Turing Machine working in that time bound?

Rephrasing does $P \neq NP$ imply that there exists some class $D$ of languages solvable by a deterministic Turing Machine, such $P \subsetneq\ D \subseteq NP$ and SAT is $D$-hard?
This post shows that the answer to your initial question is yes (even if P = NP).

On the other hand, I suspect there is no known proof that
if 
unambiguousGC(polylog, NC) ⊈ coNP/poly then there is a superpolynomial time-constructible function $T$ and a superlogarithmic space-constructible function $S$ such that

DTIME(T(n)) ∩ DSPACE(S(n)) ⊆ NP/poly.