Lossy face image compression can degrade the image quality and the utility for the purpose of face recognition. This work investigates the effect of lossy image compression on a state-of-the-art face recognition model, and on multiple face image quality assessment models. The analysis is conducted over a range of specific image target sizes. Four compression types are considered, namely JPEG, JPEG 2000, downscaled PNG, and notably the new JPEG XL format. Frontal color images from the ColorFERET database were used in a Region Of Interest (ROI) variant and a portrait variant. We primarily conclude that JPEG XL allows for superior mean and worst case face recognition performance especially at lower target sizes, below approximately 5kB for the ROI variant, while there appears to be no critical advantage among the compression types at higher target sizes. Quality assessments from modern models correlate well overall with the compression effect on face recognition performance.