Speaker-attributed automatic speech recognition (ASR) in multi-speaker environments remains a major challenge. While some approaches achieve strong performance when fine-tuned on specific domains, few systems generalize well across out-of-domain datasets. Our prior work, Diarization-Conditioned Whisper (DiCoW), leverages speaker diarization outputs as conditioning information and, with minimal fine-tuning, demonstrated strong multilingual and multi-domain performance. In this paper, we address a key limitation of DiCoW: ambiguity in Silence-Target-Non-target-Overlap (STNO) masks, where two or more fully overlapping speakers may have nearly identical conditioning despite differing transcriptions. We introduce SE-DiCoW (Self-Enrolled Diarization-Conditioned Whisper), which uses diarization output to locate an enrollment segment anywhere in the conversation where the target speaker is most active. This enrollment segment is used as fixed conditioning via cross-attention at each encoder layer. We further refine DiCoW with improved data segmentation, model initialization, and augmentation. Together, these advances yield substantial gains: SE-DiCoW reduces macro-averaged tcpWER by 52.4% relative to the original DiCoW on the EMMA MT-ASR benchmark.