That depends on what flatpak you install. It will bring in the libraries it needs for the application to function.
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It has been added and is accessible via MXPI. But if you do not install any apps with it the footprint is small.
OK, I changed that, can you please give it a test and see if it works as you wish? (going to the Output tab does reset the search field -- I think it makes sense to show all packages after you install/remove something instead of just showing the package that you searched for).
You can test this .deb: https://github.com/MX-Linux/mx-packagei ... _amd64.deb
Search term is retained though tab changes and clears once package is installed as you intended and as I believe the request intended.