The progenitor stars of long Gamma-Ray Bursts (GRBs) are thought to be Wolf-Rayet stars, which generate a massive and energetic wind. Nevertheless, about 25 percent of all GRB afterglows light curves indicate a constant density medium close to the exploding star. We explore various ways to produce this, by creating situations where the wind termination shock arrives very close to the star, as the shocked wind material has a nearly constant density. Typically, the distance between a Wolf-Rayet

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... ar and the wind termination shock is too large to allow afterglow formation in the shocked wind material. Here, we investigate possible causes allowing for a smaller distance: A high density or a high pressure in the surrounding interstellar medium (ISM), a weak Wolf-Rayet star wind, the presence of a binary companion, and fast motion of the Wolf-Rayet star relative to the ISM. We find that all four scenarios are possible in a limited parameter space, but that none of them is by itself likely to explain the large fraction of constant density afterglows. A low GRB progenitor metallicity, and a high GRB energy make the occurrence of a GRB afterglow in a constant density medium more likely. This may be consistent with constant densities beingpreferentially found for energetic, high redshift GRBs.