for the experimental observation of coherence in the mentioned time range.

Summarizing, the experimental observations indicate a strong impact of disorder on the polariton BEC even at high excitation power well above the condensation threshold. As discussed in Sec. \[sec:theory\_disorder\] the suppression of disorder effects with increasing condensate density is strongly hindered for a polariton BEC. We assume that the interplay of gain-loss and disorder prevents a stabilization at high excitation power also in the experiment.

Comparison between Theoretical Model and Experiment {#sec:fit_of_exp_data}
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In the following, we will compare our experimental observations with numerical simulations. At threshold $P={P_{{\mathrm{th}}}}$ a cross-over from a non-condensed state to a polariton BEC takes place, typically indicated by a super-linear increase of the emission intensity. Such a transition is not very well described by the used eGPE . For this reason, the data analysis is done well above threshold, where both experimentally observed and theoretically calculated blueshift (condensate density) increase linearly with pump power. We note that the evolution of the experimentally measured polariton blueshift $\Delta E$ as a function of the excitation power shows two kinks at $P = 2~P_{{\mathrm{th}}}$ and $P = 4~P_{{\mathrm{th}}}$ (cf. Fig. \[fig:Blueshift\] in Appendix \[sec:experiment\_disorderorigin\]). We believe that 