I wouldn't use this, but as long as it's optional I don't care. But Each entry would contain an "x" dismissal button which would delete the notification from both the Inbox and the dropdown sounds like an accident waiting to happen. Deleting things from the inbox is permanent, isn't it? I can see people deleting things when they meant to get rid of the dropdown, especially if it appears on the navbar without having to be configured first.
no subject