Checking if a given flow is a maximum flow I'm curious as to if there's any way to check (without having to run a 'whole' maximum-flow-algorithm) whether a given flow $f_e$ is the maximum flow of the flow graph $G$ in $O(|E|)$ time complexity.

I've thought about running one iteration of Ford-Fulkerson to see if there is an augmenting path from the source $s$ to the sink $t$, but couldn't such a solution get a time complexity worse than $O(|E|)$ in the worst case?
A flow is maximum if there is no $s$-$t$ path in the residual network. You can check this in time $O(|E|)$.