Is it true that whoever does not pray Fajr on time—meaning he makes it up after sunrise—cannot pray the Sunnah of Duha?

The Dhuha prayer is a confirmed Sunnah, proven to have been performed by the Prophet, peace and blessings be upon him. If someone misses the Fajr prayer and then makes it up after sunrise, they may pray the Dhuha prayer, for it is an independent Sunnah with no necessary connection to the Fajr prayer. It has no relation to the time of Fajr or whether Fajr was prayed on time or made up. Rather, it is prescribed for someone who has fallen short in an act or missed a good deed to increase in righteous acts, the best of which are supererogatory prayers, as stated in his, peace and blessings be upon him, saying: "Help me achieve this for yourself by prostrating much."