Access parts within AsyncDirective after setValue call

Is there any way for an AsyncDirective to access its rendered parts after a setValue call?

My directive's update method calls super.update which triggers the render method, but I need a method that's called after the setValue method call has been processed, because update is not called for that.

Are you able to hold a reference to the part within the directive class?

class MyDirective extends AsyncDirective {
  currentPart: Part;

  update(part, args) {
    this.currentPart = part;
    return this.render(...args);
  }

  render(promise) {
    promise.then((value) => {
      this.setValue(value);
      // do something with `this.currentPart` ?
    });
    return 'pending';
  }
  
}

Alternatively maybe you can add a MutationObserver to the element with the part instead so you can detect whatever change setValue has caused.