Speech enhancement employing deep neural networks (DNNs) for denoising are called deep noise suppression (DNS). During training, DNS methods are typically trained with mean squared error (MSE) type loss functions, which do not guarantee good perceptual quality. Perceptual evaluation of speech quality (PESQ) is a widely used metric for evaluating speech quality. However, the original PESQ algorithm is non-differentiable, and therefore cannot directly be used as optimization criterion for gradient-based learning. In this work, we propose an end-to-end non-intrusive PESQNet DNN to estimate the PESQ scores of the enhanced speech signal. Thus, by providing a reference-free perceptual loss, it serves as a mediator towards the DNS training, allowing to maximize the PESQ score of the enhanced speech signal. We illustrate the potential of our proposed PESQNet-mediated training on the basis of an already strong baseline DNS. As further novelty, we propose to train the DNS and the PESQNet alternatingly to keep the PESQNet up-to-date and perform well specifically for the DNS under training. Our proposed method is compared to the same DNS trained with MSE-based loss for joint denoising and dereverberation, and the Interspeech 2021 DNS Challenge baseline. Detailed analysis shows that the PESQNet mediation can further increase the DNS performance by about 0.1 PESQ points on synthetic test data and by 0.03 DNSMOS points on real test data, compared to training with the MSE-based loss. Our proposed method also outperforms the Challenge baseline by 0.2 PESQ points on synthetic test data and 0.1 DNSMOS points on real test data.