No, they don't either in principal (the application in the text is much broader than specifically al-Qaeda -- it includes al-Qaeda, the Taliban, "associated forces that are engaged in hostilities against the United States or its coalition partners", and other who commit a belligerent act against the US or its coalition partners that is seen to be "in aid to" those organizations), nor in practice, because the specific allegations justifying the detention need not be disclosed, nor even the specific detention publicly disclosed, and detentions under the act, while they permit trial by military tribunal, do not require any juridical process or access to counsel to be provided -- that's rather the point of indefinite detention -- there's essentially no practical boundary to their application (if some sympathetic third party becomes aware of the detention, its possible they might be the subject of habeas corpus proceedings, which would require some showing that there was reason to believe that the person was within the fairly broad scope in the text of the act, but that's by no means a certainty in any particular detention.)