We compared SST vs SST-CH model (Figure 1) by means of the same ABC-RF model selection framework previously adopted. The two models cannot be clearly distinguished in any of the three structured species since: i) they showed similar posterior probability (~0.50); ii) the prior error rates are large ~0.40 (Table S9); iii) posterior distributions of Nmbefore and after TCH are wide and largely overlapping (Table S10); iv) the normalized SFS closest to the observed data retrieved under the two models are very similar (Figure S6). We ran a second set of coalescent simulations focusing on the consequences of a recent change in connectivity on the observed SFS and the reconstructed stairwayplot (Table S3 and S4). The decrease in connectivity was simulated by reducing either m (the migration rate per generation) or NDEME (the effective population size of each deme). As expected, we found a signature of recent population decline in all simulated scenarios, with its intensity only slightly affected by the change in Nm (Figures 6, S7, S8 and S9). However, the drop in NDEME (Figures 6 and S9) had larger effect compared to the drop in m (Figures S7 and S8) on both the normalized SFS and the expansion time estimated by the stairwayplot. In scenarios with 100x reduction in NDEME, the stairwayplot could not retrieve the ancestral expansion even for large long-term Nm (Figure 6). FIM-CH models displayed a behaviour similar to SST-CH models but more pronounced (Figures S10, S11, S12 and S13, Table S5 and S6). While at TCH = 10 a decrease in Nm slightly affected the SFS and the reconstructed stairwayplot, the consequence of the change in connectivity are more substantial at TCH = 50, with a stronger deficit in singletons and a more pronounced recent decline in Ne particularly in scenarios with a 100-fold reduction of NDEME (Figures S11 and S13).