Profiles for logged-in users

I noticed there is a setting for "Hide my public profile". This hides it completely, for both anonymous visitors and logged in users. On the other hand, unchecking this means that the profile is publicly visible on the web and can be (e.g., indexed by search engines). Is there any way to customize the profile visibility based on whether the user is logged in?

It is already set like this:

If you go incognito, you can see this behavior: