Did the Prophet (peace be upon him) pray two rak'ahs after Fajr prayer, meaning after sunrise?

It was part of the Prophet’s guidance (peace and blessings be upon him) to remain in his prayer place after Fajr prayer until the sun rose pure and white, but there is no explicit statement that he would pray two rak'ahs after that. It has been narrated that "whoever prays Fajr in congregation, then sits remembering Allah until the sun rises, then prays two rak'ahs, will have a reward like that of a complete Hajj and Umrah," and this hadith has a good chain of transmission (hasan isnad).