You won't be able to reliably check if a user has closed a tab in MVC due to the nature of restful Http requests.
If it's absolutely essential to keep track of open tabs I would think about SignalR, you can reliably track open tabs in real time using that. You would have to implement a collection of unique session objects synced to a unique ID on the matching tab. On submitting the form you would match the session by ID to the ID passed in by the form.
Having said that I would be very cautious about keeping anything specific to a certain page/tab in a session. It can lead to many confusing bugs that are very hard to debug, often leading to complete reworks.