Performing two supererogatory Rak`ahs before Fajr Prayer suffices for Tahiyyat-ul-Masjid

Q: I know that if someone enters the Masjid (mosque) after the second Adhan (call to Prayer) of Fajr (Dawn Prayer), they are allowed to perform the Sunnah of Fajr only, not Tahiyyat-ul-Masjid (two-unit Prayer to salute the mosque). My question is: Is there any evidence that offering Tahiyyat-ul-Masjid is not allowed in this case and offering the Sunnah of Fajr would suffice? It is noteworthy that there is enough time to offer both Prayers. Appreciate your making this point clear for me, may Allah guide you! A: When the prescribed time of prayer begins and the person enters the Masjid, they have to perform the two-Rak`ah-Sunnah of Fajr with the intention of offering Tahiyyat-ul-Masjid together. The latter is incorporated into the Sunnah. On the other hand, when they perform the Sunnah at home and enter the Masjid, they need to (Part No. 6; Page No. 117

A: When the prescribed time of prayer begins and the person enters the Masjid, they have to perform the two-Rak`ah-Sunnah of Fajr with the intention of offering Tahiyyat-ul-Masjid together. The latter is incorporated into the Sunnah. On the other hand, when they perform the Sunnah at home and enter the Masjid, they need to (Part No. 6; Page No. 117) offer Tahiyyat-ul-Masjid.May Allah grant us success. May peace and blessings be upon our Prophet Muhammad, his family, and Companions.