By default the search results are displayed in an absolute position element (to allow hovering under the search bar). In this case the result box parent is the document body itself, to ensure nothing else can influence it’s position and the calculations regarding it. The reason the results stay visible is exactly because of this – there is no connection between the pop-up and the search results box.
The solution is going to be really simple, just change the Results Layout Position option to “Block” on the Layout Options -> Results layout panel: https://i.imgur.com/q6D5kvK.png
This way the results are displayed just below the search box as a block element, and stay within the scope of the pop-up box in a static position.
An alternative solution would be to trigger a document click event when the pop-up open/close button is clicked, by putting this code to the footer.php in your active theme directory:
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