If you mean the system shown in the illustration in the article, the tether's orbit does change: the illustration shows the tether's orbit changing after it cat

If you mean the system shown in the illustration in the article, the tether's orbit does change: the illustration shows the tether's orbit changing after it catches the payload and again after it releases the payload. What is weird to me is that the tether's trajectory is shown as curving upward instead of downward. That doesn't make sense to me.