Is vendor embedded AI is the antitrust battle?

The ability to switch out the AI backend provider in Co-Pilot, similar to how you can choose a preferred search engine in a browser, seems poised to become the next antitrust battleground. This situation is reminiscent of Microsoft’s approach with the ‘embedded browser’ with Internet Explorer and Windows OS.

I can see how this would apply to Microsoft, where CoPilot becomes a framework where users can specify the AI of choice to power it. Other platform providers like Apple, Android also come to mind.

I want to know why Open Ai doesn’t Sue Google just like Google sued Microsoft for embedding their browser in the operating system.
Google has embedded their Ai in their operating system and as such Gemini has a lot more functionality.
I have to use their obviously inferior Ai because others don’t have the system permissions and integration that gemini does.