Before revision:  The second self-consistent test embodies the idea that, for the lead-lag path to be significant, synchronising the two time series using the time varying lead-lag path should lead to a statistically significant correlation.
After revision:  The second self-consistent test embodies the idea that, for the lead-lag path to be significant, synchronizing the two time series using the time varying lead-lag path should lead to a statistically significant correlation.