How to Execute a scheduler task via an ajax request to an eID in typo3... TYPO3 version 4.5.16

..._eID.php

<?php

if (!defined('TYPO3_MODE')) {
    die ('Access denied.');
}

$GLOBALS['TSFE']->fe_user = tslib_eidtools::initFeUser();
tslib_eidtools::connectDB();

if(t3lib_div::_POST('wsproductlist') || t3lib_div::_GET('wsproductlist')){
  $get = t3lib_div::_POST('wsproductlist');

    if($get == '') {
        $get = t3lib_div::_GET('wsproductlist');
    }
    
    switch ($get[action]) {
      case 'executeScheduler':
            echo json_encode(executeScheduler());
            break;
    }
}

// doesn't seem to work
function executeScheduler() {
    /* @var tx_scheduler */
    $tx_scheduler = t3lib_div::makeInstance('tx_scheduler');

    $task = $tx_scheduler->fetchTasksWithCondition('classname = ..._customer', true);
    return $tx_scheduler->executeTask($task);
}
?>

script.js

...

$.ajax({
  url: 'index.php',
  type: 'POST',
  data: {
      eID: '..._eID',
      'wsproductlist\[action\]': 'executeScheduler'
  },

  // Es hat funktioniert?
  success:function (response) {
      console.log(response);
      alert('Successfully updated');
  },

  // Schade, es war ein Fehler
  error  :function (error) {
      console.log(error);
  }
});

...