My guess with the bath scene between Shinomiya and kei is that since the Shirogane mom had high hopes for kei she got put into Shuchiin since it's been said that there's kids that have been studying there for a long time and rumors about Kei's situation came up and since Shinomiya hates that kind of thing she spoke her mind to Kei which led her to moving back with her dad? Its kind of obvious but that would explain why Kei refers to Shinomiya as her favorite senpai