Add Drupal menu items with hierarchy (not working right)

add_menu_items.php

/**
 * Set the parent of menu items via name.
 *
 * @param string $menu_name
 *   Machine name of the menu.
 * @param string $item_name
 *   Title of the menu link to use as the new parent.
 * @param array $children
 *   List of menu item (mlids) to nest under parent.
 */
function _tableau_updates_set_menu_parent_by_name($menu_name, $item_name, $children) {
  // Find parent menu item.
  $menu_list = menu_load_links($menu_name);
  foreach ($menu_list as $menu_item) {
    if ($menu_item['link_title'] === $item_name) {
      // Update children items.
      foreach ($children as $child_item) {
        // Save with a new parent.
        $child_menu_item = menu_link_load($child_item);
        $child_menu_item['plid'] = $menu_item['mlid'];
        menu_link_save($child_menu_item, $child_menu_item);
      }
    }
    break;
  }
}

/**
 * Add industry sector menu items to Solutions expand-o-nav.
 */
function tableau_stories_update_7204() {
  // Intended hierarchy.
  $sectors = array(
    'Communications, Media & Technology' => array(52890, 52888, 52885, 52889),
    'Energy & Resources' => array(52816),
    'Financial Services' => array(52812, 52882, 52883),
    'Healthcare & Life Sciences' => array(52881, 52886),
    'Manufacturing' => array(52884),
    'Public Sector' => array(52811, 52814, 52815, 52880, 52817),
    'Retail & Consumer Goods' => array(52813, 52887),
    'Services' => array(),
    'Travel & Transportation' => array(),
  );
  // Create new menu items.
  $menu_properties = array(
    'plid'       => 52937, // Industries.
    'menu_name'  => 'menu-stories-topics',
    'link_path'  => '<nolink>',
    'module'     => 'menu',
    'expanded'   => 1,
    'customized' => 1,
    'language'   => 'und',
  );
  foreach ($sectors as $sector_name => $sector_items) {
    $menu_item = array_merge($menu_properties, array('link_title' => $sector_name));
    print_r($menu_item);
    menu_link_save($menu_item, array(), array(52937));
    //_tableau_updates_set_menu_parent_by_name('menu-stories-topics', $sector_name, $sector_items);
  }
  menu_cache_clear_all();
}