It's open whether $EXP = NEXP \to P = NP$ (the other direction can be shown by padding).  My question: has there been any progress along these lines at all?  For example, can we show that $EXP = NEXP \to NP \subset SUBEXP$ (where $SUBEXP = \cap_{\epsilon > 0} 2^{n^\epsilon}$)?
Upper bounds for $NP$ based on $NEXP = EXP$
complexity theory;nondeterminism