If you have Grosh Premium, you can turn on Live update in the app. This ensures your currently selected list gets updated even though it is currently closed. On iOS this includes the number shown on your Grosh icon (the so-called badge).
If this is not working, here’s a tip for you: This feature only works with one device per user, so make sure you’ve only enabled it on one device. For instance, if you have a tablet and a phone, just enable it on the device you bring with you. Typically your phone. This way, the list is updated when you bring it to a location without network coverage.