In theory, it should be possible to use localStorage/sessionStorage to do this by setting the current page URL on every page, and checking every once in a while on the "original" page to see if the value is not the same as document.location
.
If you set the current time in the storage thing as well, you can check to see if the page is closed by seeing if the "last tab time" is, say, more than 30 seconds ago, or some such.
As for the website, I visited it, and did a search, and didn't see any warnings or anything about new tabs being open at all when I tried it.