Apple AI & Siri respond with Google search even though I have specified DuckDuckGo as my search engine

iPhone 16 pro max iOS 18.3.1. I have specified DuckDuckGo as my search engine. Yet Apple AI & Siri search Google and respond with Google search. I absolutely avoid Google with its “lack” of privacy. How can I cause Apple AI & Siri to use DuckDuckGo? I have turned Apple AI off subsequently. As a side note Siri with Apple AI on was worse at everything than Siri alone.

Any ideas on how to overcome or remediate these issues? Thank you.