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Multiple sclerosis lesion activity segmentation is the task of detecting new and enlarging lesions that appeared between a baseline and a follow-up brain MRI scan. While deep learning methods for single-scan lesion segmentation are common, deep learning approaches for lesion activity have only been proposed recently. Here, a two-path architecture processes two 3D MRI volumes from two time points. In this work, we investigate whether extending this problem to full 4D deep learning using a history of MRI volumes and thus an extended baseline can improve performance. For this purpose, we design a recurrent multi-encoder-decoder architecture for processing 4D data. We find that adding more temporal information is beneficial and our proposed architecture outperforms previous approaches with a lesion-wise true positive rate of 0.84 at a lesion-wise false positive rate of 0.19.