"Updating Your Progress" -> this is how you kill the checkmark
This has been working for me lately. I'm playing on an iPad Mini. It may not work for others.
1. start the app
2. if you get the "Updating Your Progress" message, just hard close out of the app.
DO NOT CLICK THE CHECKMARK.
3. Restart the app
4. As soon as you're past the loading screen and hit the main map, start jamming your finger down on the "AstroTrain Button" (on the bottom right corner). This should launch you into the Barracks.
5. The "Updating Your Progress" pop-up will still come up. That's OK, because as soon as the Barracks screen loads, the "Updating Your Progress" pop-up will go away.
6. Exit back onto the map, and voilà! No more "Updating Your Progress".
7. Enjoy playing the game.
Hope this helps. Unfortunately it's not a permanent fix, but it's good for as long as you don't close the game. But then you can just follow steps (1) to (7) again.