Surely the data must go to the OpenAI servers, how else would they use LLMs on it? We cannot see if that data ends up in the training data.

Personally I would just believe what they say for the time being; there would be backlash in doing something else, possibly legal one.

I think the original claim was about something different. "Is it confirmed that site loads..." - I read it as the author taking about general browsing, not just explicit questions, with the context of the page.

Whatever is included in context is in OpenAI's control from that point forward, and you just have to trust them not to do anything with it.

That isn't a conspiracy theory, it's fundamentally how interfacing with 3rd party hosted LLMs works.