When I followed the steps on your doc page,
(https://documentation.ajaxsearchpro.com/elementor-integration/elementor-pro-product-widget-live-filter|)
the Product widget does not update when I filter using Search Bar and/or check boxes with Product Categories
See attached screenshots of test page Select-all and Select-None to see that the Product widget does not update when the check box filter is changed.
It is hard to tell only by screenshots, but my guess is the missing asp_es_{search_ID} class from the products widget. Check this part of the linked documentation. It is a special CSS class name (different for every search instance), which has to be applied to the Products widget on the advanced options configuration.