it won't. My bet is that bing and some other indexes are 95% Ok for average Joe. But relevance ranking is much tougher problem, and "google.com" is household brand with many other functions(maps, news, stocks, weather, knowledge graph, shopping, videos), and that's what is foundation of google monopoly.
I think this shared index thing will actually kill competition even more, since every players will use only index owned by google now.
This was 10 years ago. I could argue a moral Superior that Google possessed over Microsoft and Facebook, but man those days are looooooong gone.
I mean, they're still going to be the number 1 name in adtech and analytics. And they're still gonna have pretty decent personalized ads because of analytics.
Plus, that just one part of their business. There's also Android, which is a money printing machine with the Google Store (although that's under attack too).