PHP set private properties without setters statically

private_property_test.php

<?php

class User
{
    private $name;

    public function __construct()
    {
    }

    public static function register(string $name)
    {
        // Create a new class
        $user = new self();

        // Set private property ???
        $user->name = $name;

        // Return class
        return $user;
    }

    public function getName()
    {
        return $this->name;
    }
}

// Works ???

$user = User::register('username');
var_dump($user->getName()); // username

// Not working as expected

$user = new User();
$user->name = 'username'; // Error: Cannot access private property User::$name