This paper evaluates the extent to which current Large Language Models (LLMs) can capture task-oriented multi-party conversations (MPCs). We have recorded and transcribed 29 MPCs between patients, their companions, and a social robot in a hospital. We then annotated this corpus for multi-party goal-tracking and intent-slot recognition. People share goals, answer each other's goals, and provide other people's goals in MPCs - none of which occur in dyadic interactions. To understand user goals in MPCs, we compared three methods in zero-shot and few-shot settings: we fine-tuned T5, created pre-training tasks to train DialogLM using LED, and employed prompt engineering techniques with GPT-3.5-turbo, to determine which approach can complete this novel task with limited data. GPT-3.5-turbo significantly outperformed the others in a few-shot setting. The `reasoning' style prompt, when given 7% of the corpus as example annotated conversations, was the best performing method. It correctly annotated 62.32% of the goal tracking MPCs, and 69.57% of the intent-slot recognition MPCs. A `story' style prompt increased model hallucination, which could be detrimental if deployed in safety-critical settings. We conclude that multi-party conversations still challenge state-of-the-art LLMs.