I have a game I downloaded from Apple Arcade on my Apple TV. (The game in question is Leo's Fortune, but I am making the question as app agnostic as possible)

When i first downloaded and played the game, I was signed in to my iCloud user on the Apple TV (long press home button, selected my profile)

Now, any other family members signed in to the Apple TV using their iCloud accounts all see my save data for this Arcade game — even if they sign in; even if I delete the app entirely from the Apple TV and then they download it.

Other games work properly (Angry Birds Reloaded, as one example)

How can I correct a situation where one Apple Arcade game doesn't allow me to change users, always showing the same save data regardless of what I have tried?