It's not war, it's indifference. Why should Docs give a damn what Chrome's priorities are? The only reason they would is if there were effective leadership ensuring company-wide alignment, but there is no one filling that function.
Because if they did company wide alignment, you would kill their promotion pipeline. Want harmony? Then in Google you build your own app (see N+1 chat apps) to get that feature, get promoted and dump the project on someone else.
Yeah, the classic image still holds true.
https://www.globalnerdy.com/2011/07/03/org-charts-of-the-big...