GlFilteredSearch: Date filter with datepicker

I would like to use especially the filtered-search component. Here I have already created and tried some tokens. Now I would like to add a date filter inside the searchbar.
Is there a way to add a datepicker/ daterangepicker inside the filter input field? That after the operator a datepicker is ready for date selection and add the selected value as filter criteria?
Or can I add a datepicker component in the suggestion list?
I would be very grateful for any ideas or help