Is NP in $DTIME(n^{poly\log n})$?
$DTIME(n^{polylogn})$ is known as $QP$ (quasi-polynomial).

It is widely believed that $NP\not \subset QP$, although it is a stronger statement than $P\neq NP$.

Some common conjectures, such as the  Exponential Time Hypothesis imply $NP\not \subset QP$.