GAN-based image restoration inverts the generative process to repair images corrupted by known degradations. Existing unsupervised methods must carefully be tuned for each task and degradation level. In this work, we make StyleGAN image restoration robust: a single set of hyperparameters works across a wide range of degradation levels. This makes it possible to handle combinations of several degradations, without the need to retune. Our proposed approach relies on a 3-phase progressive latent space extension and a conservative optimizer, which avoids the need for any additional regularization terms. Extensive experiments demonstrate robustness on inpainting, upsampling, denoising, and deartifacting at varying degradations levels, outperforming other StyleGAN-based inversion techniques. Our approach also favorably compares to diffusion-based restoration by yielding much more realistic inversion results. Code will be released upon publication.