If you are adding posts from a front end submissions tool, that tool must make a call to our plugin to tell it that the posts has changed. See this action:

https://searchandfilter.com/documentation/action-filter-reference/#update-cache-for-a-particular-post

This will trigger the same action in our plugin as though you had pressed the Update post button in the editor.

The problem is arising because the script you are using is probably not using the WordPress save_post() function.

There is not, sorry. What import tool/plugin are you using?

See this post (it might work):if using WP All Import, have you have included this code in functions.php?

add_action('pmxi_saved_post', 'wp_all_import_post_saved', 10, 1);

function wp_all_import_post_saved($id) {
    do_action('search_filter_update_post_cache', $id);
}

As this (should) update our cache each time you use that plugin.