Did the Prophet (peace be upon him) enter the the mosque with his right foot and exit with his left, despite the weakness of the chain of transmission of the hadith mentioned in that regard?

The ruling on entering the mosque with the right foot and exiting with the left foot. It is mentioned that Al-Hakim authenticated the hadith of Anas ibn Malik regarding this, while Al-Bayhaqi deemed it weak, and Al-Albani classified it as good (hasan). The desirability of this act is inferred from Aishah's hadith about the Prophet's (peace be upon him) love for beginning with the right in all his affairs, and from the sayings of jurists and scholars of Usul (principles of Islamic jurisprudence) that the statement of a Companion: "It is from the Sunnah" has the ruling of being مرفوع (traced back) to the Prophet (peace be upon him). It is also emphasized that certainty is not a condition for evidence, and rulings are established by probabilistic (dhanni) evidence.