What studies isolate back-and-forth LLM interaction from one-way sharing and self-refinement [D]

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I'm trying to find out whether back-and-forth between two different LLMs improves task success beyond strong alternatives under a controlled resource budget — and I'd rather hear that it's already settled than spend money finding out. AI-assisted throughout: the protocol was developed and critiqued with ChatGPT and Claude. Their citation handling turned out unreliable, which is why I'm asking for sources rather than citing any. The literature check is incomplete and I'm making no novelty claim.