Drupal 8 - MenuLinkContent get all children via loadByProperties

I need help to get all MenuLinkContent children from a specific MenuLinkContent and can't find a solution. I tried several different ways, but no success at all.

Here goes my code:

//get all MenuLinkContent published with name 'myname'
$main_menu = \Drupal::entityTypeManager()->getStorage('menu_link_content')
      ->loadByProperties(['menu_name' => 'myname' , 'enabled' => 1]);

foreach ($main_menu as $menu) {
  //could not find a better solution, so i have to check if parent is empty.
  if ($menu->getParentId()=='') {
    //here i'm triyng to get all children, 
    $child_menu = \Drupal::entityTypeManager()->getStorage('menu_link_content')
          ->loadByProperties(['menu_name' => 'myname',  'parent' => $menu ]); 

As you can see the property 'parent' do not fit with '$menu'. I tried several different properties from '$menu' and no one seems to match my query.

If more information is needed, just ask and i will post here. Other ways to achieve this iteration are welcome too.

Thanks in advance.

Got it, found this code from @Slim and works fine:

Below code was taken from this answer https://drupal.stackexchange.com/a/224786/89808

"I'm quite late, but maybe helps someone looking for answers, here's my solution for generating recursive array from menu items." @Slim

private function generateSubMenuTree(&$output, &$input, $parent = FALSE) {
$input = array_values($input);
foreach($input as $key => $item) {
  //If menu element disabled skip this branch
  if ($item->link->isEnabled()) {
    $key = 'submenu-' . $key;
    $name = $item->link->getTitle();
    $url = $item->link->getUrlObject();
    $url_string = $url->toString();

    //If not root element, add as child
    if ($parent === FALSE) {
      $output[$key] = [
        'name' => $name,
        'tid' => $key,
        'url_str' => $url_string
      ];
    } else {
      $parent = 'submenu-' . $parent;
      $output['child'][$key] = [
        'name' => $name,
        'tid' => $key,
        'url_str' => $url_string
      ];
    }

    if ($item->hasChildren) {
      if ($item->depth == 1) {
        $this->generateSubMenuTree($output[$key], $item->subtree, $key);
      } else {
        $this->generateSubMenuTree($output['child'][$key], $item->subtree, $key);
      }
    }
  }
}

And call that function with

//Get drupal menu
$sub_nav = \Drupal::menuTree()->load('sub-navigation', new \Drupal\Core\Menu\MenuTreeParameters());
//Generate array
$this->generateSubMenuTree($menu_tree2, $sub_nav);