Restricted product displaying in search results

Hi there!

I have a range of products listed in our e-com store that are restricted by specific user IDs. To do this, we use a plugin called Members Only.

These products are private and cannot be purchased or viewed by the general public – only specific logged in users can see their specific custom products. To a general user browsing our website, these products do not display in the shop (which is correct). However, if someone was to use the search bar for an item (e.g. “Flag”), these private, restricted products display in the search dropdown.

Search result dropdown, logged out user:
https://www.dropbox.com/scl/fi/590zcjp3uqnwnlsk8oro9/ClientPortal_LoggedOut_Flag.png?rlkey=xvg6qvjiuxf7o6w0tinu7s0tq&dl=0

When you click “view all results”, these products don’t display in the results page – so it’s only in the dropdown itself.

These products are set to not display in the search results, as a failsafe:
https://www.dropbox.com/scl/fi/0fjw5836a5tbzzuwky6hn/ClientPortal_SuperButcherFlag_ShopOnly.png?rlkey=mnxq6izhmmw1x4ra5y3c1bbd9&dl=0

Yet they’re still displaying.

I’ve reached out to Plugin Republic, the dev for Members Only, and they’ve said it’s a theme issue:

https://www.dropbox.com/scl/fi/lt25qg849jq9r4onnvntl/ClientPortal_PluginRepublicReply.png?rlkey=iw9xyh1dfv3dq4muzqfmhmmtk&dl=0

Is there a filter they could use to check if the product is restricted before it gets displayed in the search results dropdown please?

Thank you in advance for your help :)