Problems in NC not known to lie in NC2 Are there interesting problems that are in $\mathsf{NC}$ but not known to be in $\mathsf{NC^{2}}$? In the paper 'A Taxonomy of Problems With Fast Parallel Algorithms', Cook mentions that MIS was known to only be in $\mathsf{NC^{5}}$ but this has since been brought down to $\mathsf{NC^{2}}$. I am wondering if there are any other problems with polylog-depth parallel algorithms where we seem to be stuck on improving the depth. 

To narrow down even further, are there any problems in $\mathsf{NC^{2}}$ that are not known to be in $\mathsf{AC^{1}}$ or $\mathsf{DET}$?
Disclaimer: I'm not an expert in fast parallel algorithms, hence the probability that I missed more recent results that put the problems I mention in lower levels of the $\mathsf{NC}$ hierarchy is non-negligible. If you observe that it is the case, please tell me and I'll update my answer.


The report Parallel Algorithms for Depth-First Search discusses known parallel algorithms for DFS on various types of graphs. The list given on pages 9-10 indicates several algorithms in $\mathsf{NC} \setminus \mathsf{NC}_2$, such as DFS for planar undirected graphs, or in $\mathsf{RNC} \setminus \mathsf{RNC}_2$, such as DFS for general undirected graphs.
With a quick search, I could not find papers improving over the parallel algorithms for sparse multivariate polynomial interpolation over finite fields of this paper, which is in $\mathsf{NC}_3$. However, several papers that could possibly have been relevant were behind a paywall.
Computing all maximal cliques in a graph is in $\mathsf{NC} \setminus \mathsf{NC}_2$ when the number of maximal clique is polynomially bounded, according to this paper.
The maximal path problem seems to be in $\mathsf{NC}_5$ for general (undirected) graphs, I've not found a faster parallel algorithms without restrictions on the underlying graph.


Other potential candidates might include algorithms for finding perfect matchings in specific types of graphs, or algorithms for finding a maximal tree cover in arbitrary graphs (e.g. this paper mentions a randomized polytime algorithms in parallel time $O(\log^6n)$). This paper also mention solving classes of CSPs problems that arise in computer vision application, in parallel time $O(\log^3n)$.