I don't think Anthropic and OpenAI will IPO at all. Word is that Anthropic will have $100 BN in revenues this year, and very likely OpenAI will get some similar amount. You IPO when you need money, and I think Anthropic and OpenAI are past that point.
That level of revenue is apparently not enough for them to feel confident in the stability of their competitive position.
You also IPO if your investors want to cash out.
They want to spend over $200B/year on $100B of revenue so they still need investment.