As soon as you are disconnected from the internet, Supernotes will enter a read-only mode. You can see if this is the case because your user icon (top right) will have a red dot beside it if you are disconnected from the Supernotes servers. You will still be able to browse, read and search your notes, like so:
Unfortunately while offline you will not be able to edit any cards. That being said, a fully offline experience is at the top of our priority list and should be available soon.