Conversational breakdowns in close relationships are deeply shaped by personal histories and emotional context, yet most NLP research treats conflict detection as a general task, overlooking the relational dynamics that influence how messages are perceived. In this work, we leverage nonviolent communication (NVC) theory to evaluate LLMs in detecting conversational breakdowns and assessing how relationship backstory influences both human and model perception of conflicts. Given the sensitivity and scarcity of real-world datasets featuring conflict between familiar social partners with rich personal backstories, we contribute the PersonaConflicts Corpus, a dataset of N=5,772 naturalistic simulated dialogues spanning diverse conflict scenarios between friends, family members, and romantic partners. Through a controlled human study, we annotate a subset of dialogues and obtain fine-grained labels of communication breakdown types on individual turns, and assess the impact of backstory on human and model perception of conflict in conversation. We find that the polarity of relationship backstories significantly shifted human perception of communication breakdowns and impressions of the social partners, yet models struggle to meaningfully leverage those backstories in the detection task. Additionally, we find that models consistently overestimate how positively a message will make a listener feel. Our findings underscore the critical role of personalization to relationship contexts in enabling LLMs to serve as effective mediators in human communication for authentic connection.