iCloud+ Custom Domain email not showing new email badges on iOS

I've successfully set up an iCloud custom domain email address. As this custom address shares the inbox with the base icloud.com address, on iCloud web I created a rule where all new emails received to the custom address are moved to a new 'inbox' folder. I then placed this folder as a 'favorite mailbox'.

When I receive an email to the custom email address on Mac, I then receive a notification for this email and a 'new email' badge on the Mail app. On iOS, on the other hand, there's no badge on the iOS Mail app. I do get a badge for emails sent to the regular icloud account, but not on the custom email address.

Is there a way to make this 'inbox folder' also trigger badge count? Or is there a better way to organize the inbox so I get iCloud and custom domain emails separate from each other? In iOS Mail settings, 'Customize Notifications' badges are ON for the iCloud account, but Favorite Mailboxes only allows to activate Alerts and Sounds.

Thank you