Open question regarding the DFS It is an open question that whether $\mathsf{DFS\text{}}$ can be done in $O(m+n)$ time and $O(n)$ bits of space, where $n,m$ represent the number of vertices and edges in a undirected graph. see this for DFS

DFS for directed graph is known to be in RNC i.e. randomised NC. $\mathsf{DFS\text{}}$ also known to be $\mathsf{P\text{-}Complete}$ problem.

My question is what would happen if someone shows that  $\mathsf{DFS\text{}}$ can be done in $O(n+m)$ time and in $O(n)$ bits of space?Is it likely in that situation for DFS to admit polyloga-rithmically fast, parallel algorithm using only polynomial resources (see this research paper)?
It is conjectured that $\mathsf{P} \neq \mathsf{NC}$, and so no $\mathsf{P}$-complete problem has polylogarithmic parallel algorithms. This in no way contradicts the existence of a linear time and space algorithm for DFS.