The popular 2D Fisher-KPP equation with constant parameters fails to predict activated or inhibited cell-sheet wound closure. Here, we consider the case where the collective diffusion coefficient is time dependent, with a 3-parameter sigmoid profile. The sigmoid is taken S-shaped for the activated assays, and Z-shaped for the inhibited ones. For two activated and two inhibited assays, our model is able to predict with a very good accuracy features of the wound closure like as the time evolution of the wound area and migration rate. The calibrated parameters are consistent with respect to different subsets of the experimental datasets used for the calibration. However, the assumption of sigmoid time profile for the proliferation rate yields calibrated parameters critically dependent on the dataset used for calibration.