  We show that the use of a recently proposed iterative collision model with
inter-environment swaps displays a signature of strongly non-Markovian dynamics
that is highly dependent on the establishment of system-environment
correlations. Two models are investigated; one in which such correlations are
cancelled iteratively and one in which they are kept all across the dynamics.
The degree of non-Markovianity, quantified using a measure based on the trace
distance, is found to be much greater for all coupling strengths, when
system-environment correlations are maintained.
