$P \neq NP$ and determinism
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.