Developing robust speaker verification (SV) systems without speaker labels has been a longstanding challenge. Earlier research has highlighted a considerable performance gap between self-supervised and fully supervised approaches. In this paper, we enhance the non-contrastive self-supervised framework, Self-Distillation Prototypes Network (SDPN), by introducing dimension regularization that explicitly addresses the collapse problem through the application of regularization terms to speaker embeddings. Moreover, we integrate score normalization techniques from fully supervised SV to further bridge the gap toward supervised verification performance. SDPN with dimension regularization and score normalization sets a new state-of-the-art on the VoxCeleb1 speaker verification evaluation benchmark, achieving Equal Error Rate 1.29%, 1.60%, and 2.80% for trial VoxCeleb1-{O,E,H} respectively. These results demonstrate relative improvements of 28.3%, 19.6%, and 22.6% over the current best self-supervised methods, thereby advancing the frontiers of SV technology.