Unfortunately... you literally can't.

I have browser extensions that block the AI searches everywhere (Google, DDG, etc), and whenever there's a lag, I notice that the AI search still auto-populates at the front of the page regardless before the extension hides it.

It's always running, whether you consent or not. I mean, I guess you could just stop searching for information on the internet? But that's not really inconveniencing anyone but yourself.