I’m afraid it is not the plugin causing the window closing there, as the popup is not controlled by the plugin either. I tested by removing the search from the client side, and replacing it with an empty input field. The pop-up closes whenever the input is focused, regarless it is the search input or not: https://youtu.be/vgh1uc2wSCY
That means the issue is unrelated to the search plugin, and it is probably an issue with the pop-up script.