Payment Gateway conditions don't work properly for shipping methods in checkout

I followed the idea from #4 (thank you lubwn) and modified it accordingly to keep any currently selected payment method:

(function ($) {
  /**
   * Refresh the payment information about updating the shipping information.
   *
   * @see https://www.drupal.org/project/commerce_shipping/issues/3226851#comment-14761180
   */
  $(document).ajaxComplete(function(event, xhr, settings) {
    if (settings.extraData !== undefined && settings.extraData._triggering_element_name.indexOf( "shipping_information") !== -1) {
      let element = $('[id^="edit-payment-information-payment-method"]');
      element
        .find('input:checked').prop("checked", false)
        .click();
    }
  });
})(jQuery)

Here is a patch to update the selected payment gateway using #ajax callback.
It modifies the ShippingInformation::buildPaneForm() to set the payment_gateway order field when possible. This must be done before building the form and calculating the available shipments.

The patch should be further improved before committing to be more general on form IDs but fixed the issue for me.