Natural candidates for NP-E and E-NP

It has been known since the early 70's that $${\bf NP}$$ and $${\bf E}=DTIME(2^{O(n)})$$ are not equal (because $${\bf E}$$ is not closed under polynomial-time many-one reductions, in contrast to $${\bf NP}$$). As far as I know, however, it is still open whether one class is a subset of the other, or they are incomparable, meaning that $${\bf NP}-{\bf E}$$ and $${\bf E}-{\bf NP}$$ are both nonempty.

Question: Which are some (preferably natural) problems that are candidates for being in $${\bf NP}-{\bf E}$$ or $${\bf E}-{\bf NP}$$, assuming the respective set is not empty? I am particularity interested in natural problems within $${\bf NP}$$ that likely require exponential time with superlinear exponent, i.e., they are in $${\bf NP}-{\bf E}$$.

$$L = \{ C \ |\$$Circuit $$C$$ describes a graph $$G$$ on $$|C|^5$$ nodes where $$G$$ is 3-colorable$$\}$$.
$$L$$ is in NP, unlikely to be in $$E$$ but not that natural.