There's a very similar FOSS alternative that can be self-hosted called Perplexica. You know what's really funny?
When you ask Perplexity "Perplexity vs Perplexica", it returns this answer:
Recent evaluations indicate that Perplexica often outperforms Perplexity AI in terms of interaction quality and content relevance. In various tests, Perplexica scored higher in comprehensiveness and user-friendly interactions compared to Perplexity's different models (Vanilla and Copilot)
and
In summary, while both Perplexica and Perplexity AI aim to improve the search experience through AI technologies, Perplexica stands out as a more flexible and privacy-focused alternative. Its open-source nature allows for greater customization and control, making it suitable for users who prioritize data privacy and adaptability in their search tools.
LOL
Perplexity just acknowledged that it's worse than its open source competitor