There are widespread fears that conversational AI could soon exert unprecedented influence over human beliefs. Here, in three large-scale experiments (N=76,977), we deployed 19 LLMs-including some post-trained explicitly for persuasion-to evaluate their persuasiveness on 707 political issues. We then checked the factual accuracy of 466,769 resulting LLM claims. Contrary to popular concerns, we show that the persuasive power of current and near-future AI is likely to stem more from post-traini...