While task-agnostic debiasing provides notable generalizability and reduced reliance on downstream data, its impact on language modeling ability and the risk of relearning social biases from downstream task-specific data remain as the two most significant challenges when debiasing Pretrained Language Models (PLMs). The impact on language modeling ability can be alleviated given a high-quality and long-contextualized debiasing corpus, but there remains a deficiency in understanding the specifics of relearning biases. We empirically ascertain that the effectiveness of task-agnostic debiasing hinges on the quantitative bias level of both the task-specific data used for downstream applications and the debiased model. We empirically show that the lower bound of the bias level of the downstream fine-tuned model can be approximated by the bias level of the debiased model, in most practical cases. To gain more in-depth understanding about how the parameters of PLMs change during fine-tuning due to the forgetting issue of PLMs, we propose a novel framework which can Propagate Socially-fair Debiasing to Downstream Fine-tuning, ProSocialTuning. Our proposed framework can push the fine-tuned model to approach the bias lower bound during downstream fine-tuning, indicating that the ineffectiveness of debiasing can be alleviated by overcoming the forgetting issue through regularizing successfully debiased attention heads based on the PLMs’ bias levels from stages of pretraining and debiasing.