It's hard to say what's going to happen, but i think it might cut a decent chunk of Google users. Don't forget that one of Google's top competitors, Microsoft, is the biggest investor in OpenAI.
Apple has also partnered with OpenAI recently, and imagine if Apple at some point decides to stick SearchGPT in iPhones by default...
Check how Alphabet is dropping, especially after OpenAI announced the search engine launch:
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As soon as i was about to post this message, a notification arrived to get Gemini. No, i don't need it—I don't want bots messing about on my phone, lol.