Digital pathology offers a groundbreaking opportunity to transform clinical practice in histopathological image analysis, yet faces a significant hurdle: the substantial file sizes of pathological whole slide images (WSIs). Whereas current digital pathology solutions rely on lossy JPEG compression to address this issue, lossy compression can introduce color and texture disparities, potentially impacting clinical decision-making. Whereas prior research addresses perceptual image quality and downstream performance independently of each other, we jointly evaluate compression schemes for perceptual and downstream task quality on four different datasets. In addition, we collect an initially uncompressed dataset for an unbiased perceptual evaluation of compression schemes. Our results show that deep learning models fine-tuned for perceptual quality outperform conventional compression schemes like JPEG-XL or WebP for further compression of WSI. However, they exhibit a significant bias towards the compression artifacts present in the training data and struggle to generalize across various compression schemes. We introduce a novel evaluation metric based on feature similarity between original files and compressed files that aligns very well with the actual downstream performance on the compressed WSI. Our metric allows for a general and standardized evaluation of lossy compression schemes and mitigates the requirement to independently assess different downstream tasks. Our study provides novel insights for the assessment of lossy compression schemes for WSI and encourages a unified evaluation of lossy compression schemes to accelerate the clinical uptake of digital pathology.