Duplicating task templates when duplicating a project via API?

I’m utilizing the duplicate API call to copy a project. However, it doesn’t also copy the new “task templates” to the newly created project.

Here is some of the data I am telling it to include during the duplication. Does the API not yet have the option to include the task templates? When I Duplicate the project manually, within Asana itself, they come over fine. How do I get them to come over via the API?

var taskTitle     = inputData.taskName;
var projectID     = "123456";
var teamID        = "9999";
let body = {
    "data": {
        "include": [
            "notes",
            "members",
            "task_notes",
            "task_subtasks",
            "task_attachments",
            "task_dependencies",
            "task_projects"
        ],
        // set the Project Name text value to be the this value from the above variable
        "name": taskTitle,
        // set the Team ID value to be the this value from the above variable
        "team": teamID
    }
};

It does not, sadly.