When users take action on an action page within a site that is not the primary site of the nation, they are not seen as 'sorta-logged-in' and are instead seen as completely logged out. The sorta-logged-in status is used when the system thinks they know who the user is but cannot be sure, as they have not logged-in with a password (or social media logins). That's where it might say "Hi Brian! (Not Brian? Click here to sign in)".
Thanks for bringing this issue to our attention. We don't have a timeline for a fix at the moment but we will keep you updated here!