That's a great clue. I think what is happening is that you're experiencing a long-standing quirk of Prodigy, where closing/reopening the tab causes examples to get missed by the user. Luckily when this happens the examples aren't gone forever, just until you restart the annotation task (at which point they end up in the queue again.)
Prodigy streams don't have any mechanism built-in for returning unanswered questions, which means if you have items on the client that are not returned with answers, the server doesn't know and will give you questions starting after the last one it previously sent you when you ask again. You can see the effect here: https://giphy.com/gifs/SvcmLM0NPixm5vTapk
Do you think this could account for the issues you have seen? I can see how this is not great for a use-case with restrictions like yours. I'm looking into an enhancement that will work with your requirements and will follow up here when I'm done with that.
If what I've described is your problem, in the short-term you can ask your developers to keep the tab open until they are done with the task.