Eccentricity damping efficiency versus gap-depth for all setups where no vortex emerged. In both panels, the e-damping efficiency on the vertical axis is normalised by the expected value from Cresswell & Nelson (2008). A 20% error around the expected value in the limit of no gap (to the right in the plots) is shown by two dashed horizontal grey lines. Values for the e-damping efficiency are shown for different eccentricities, e/h ∈ {0.2, 0.4, 0.6, 0.8, 1), by points of different colours joined together by opaque lines. Panel a shows on the horizontal axis the gap depth according to the prediction of Kanagawa et al. (2018), which results in an extremely noisy scatter plot. Panel b shows, on the horizontal axis, the gap depth observed from the actual simulations as plotted in Fig. 2, which shows a much cleaner dependence. In both panels, the overall trend clearly shows a decrease in e-damping efficiency (i.e. longer e-damping timescales) for deeper and deeper gaps, down to a factor of ~l/4 less efficient eccentricity damping at the transition from type-I to type-II regimes (gap depths of ≃0.25) as compared to the limit of no gap.