Did Adam, peace be upon him, upon his expulsion from Paradise, see "Lā ilāha illā Allāh Muhammad Rasūl Allāh" (There is no god but Allah, Muhammad is the Messenger of Allah) written on its gate, and what is the authenticity of this statement and who among the scholars narrated it?

It has been reported in weak narrations, with different wordings, from Al-Tabarani, Al-Hakim, and others, that Adam supplicated to God through Muhammad after his sin, and he recognized Muhammad from the inscription "Lā ilāha illā Allāh Muhammad Rasūl Allāh" (There is no god but Allah, Muhammad is the Messenger of Allah) on the pillars of the Throne. God then responded to him that Muhammad was the dearest of creation to Him, and that He forgave Adam because of Muhammad's blessing, and had it not been for him, Adam would not have been created. Sheikh Al-Albani described this hadith as fabricated (mawdu').