Yes, the model is not lossless as this would require learning the PDF in the original input space. However, the model does learn a conditional probability dist

Yes, the model is not lossless as this would require learning the PDF in the original input space. However, the model does learn a conditional probability distribution over a lower-dimensional representation of the original image - this is unavoidable as entropy coding requires a distribution over discrete symbols. The GAN is almost auxiliary and not a central component of the model - in fact, you can get very good results without the GAN, but does seem to result in visually superior reconstructions.