What is the ruling on reciting the supplication for completing the Qur'an while sitting?

Supplicating after completing the Qur'an is desirable and مشروع (legitimate/prescribed), but there is no evidence in the Qur'an or Sunnah for its legitimacy within the prayer after completing the Qur'an. Sheikh Ibn 'Uthaymeen mentioned that he does not know of any basis in the Sunnah for completing the Qur'an within the prayer, and that the most that has been reported is the action of Anas ibn Malik (may Allah be pleased with him) gathering his family to supplicate upon completion.