For simulations we used random-sequential update with fast hopping probability $p=1$. If not specified else, we fix $q=0.6$ as slow hopping rate. The unit of time is just one timestep so that probabilities and rates have the same numerical value.

Few defects/vanishing fraction of defects
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Before we consider finite defect densities $\phi>0$ we discuss systems with a fixed number of defects in the continuum limit ($\phi=0$). Figs. \[LD-profs\]–\[S-profs\] display the dependence of the densities and the current on the position in the system.

Fig. \[LD-profs\] shows the density and current profiles of a TASEP/LK-system with five defects, a homogeneous TASEP/LK-system and a TASEP with five defects in the low density phase. The density profiles of inhomogeneous and homogeneous TASEP/LK-systems differ only in the occurrence of narrow density peaks at the defects, while globally the density profile is the same. The current profiles of the homogeneous and inhomogeneous system are identical. In contrast, the density profile of the TASEP with defects at the same sites shows density peaks as well, but the current profile (and the density profile far from the boundaries) is flat. This is due to particle conservation while the lateral influx of particles allows a spatial variation 