No, they will not. And that's because:
* A large fraction of people are realizing that some search engines are soft-censoring, already;
* Another fraction of people will not accept AI agent slop as a replacement for website search;
* Another fraction of users will get annoyed/tired with not getting directed anywhere;
* Another fraction of users does not rely on AI-focused/AI-exclusive search engine.
Between the lot of those, the non-Google-covered Internet has, and will, live on. Yes, with _less_ traffic - nto _no_ traffic.