No, I do not have studies. Tokenadult
https://news.ycombinator.com/user?id=tokenadult might have something like that. He worked in public schools for many years and ran/runs the website Learn in Freedom, one of the early pro homeschooling sites (
http://learninfreedom.org/).
My knowledge of this is rooted in having been involved in the online gifted homeschoolers community. I was briefly Director of Community Life for The Tag Project. http://www.tagfam.org/ So, you can take my word for it as a SME or you can contact Tokenadult and see if he can hook you up.
I do think your assumptions are in error. Good socialization is not about a wide variety of superficial relationships. It is about having positive interactions that teach one what one needs to know. I do know that studies show that kids who go off to boarding school do best if they figure out how to artificially replace the natural family unit with a close group of caring friends. But I don't have links to anything. That is just remembered off the top of my head.
Best.