In the neutral zone between polynomial and sub-exponential What are examples of problems that are known to be sub-exponential, but

are known to be non-polynomial, or
are not known to be polynomial?

EDIT. Here is what I mean by sub-exponential (apologies for leaving this open). A function $f:\mathbb{N}\to\mathbb{N}$ is subexponential if $$f(n) = 2^{n^{o(1)}}.$$ (Thanks to Emil Jerabek for his help in the comment section below!)
Graph Isomorphism is known to be in quasipolynomial time but not known to be in polynomial time: https://arxiv.org/abs/1512.03547
You can also come up with synthetic problems by the time hierarchy theorem, which essentially states that there are problems at any (deterministic) time complexity but not below: https://en.wikipedia.org/wiki/Time_hierarchy_theorem