Hi,
Thank you for the details, it helps a lot!
I think the only possible solution here is to use a custom script to remove the tab indexes from the search input. Try adding this code to the functions.php file in your theme/child theme directory – make sure to have a full server back-up first for safety. For more details you can check the safe coding guidelines.
add_action('wp_footer', 'asp_custom_footer_script', 999);
function asp_custom_footer_script() {
?>
<script>
jQuery(function($){
$('.asp_w *[tabindex]').removeAttr('tabindex')
});
</script>
<?php
}I will try to figure out a better way of handling this in the upcoming release.