What is the ruling regarding a mahram's refusal to accompany his female relative on her journey, despite his ability to do so and her covering his expenses, and his claim that "the world is safe and a mahram is not necessary for her"?

The Prophet, peace and blessings be upon him, forbade women from traveling without a mahram to prevent fitan (trials/temptations) and to safeguard them. This prohibition includes all journeys. The woman's guardian is responsible for her and must prevent her from traveling without a mahram, for he is a shepherd and responsible for his flock. The mahram must travel with the woman if he has no excuse and should not neglect her interests; rather, he is commanded to do so, just as the Prophet, peace and blessings be upon him, commanded a man to abandon Jihad and travel with his wife for Hajj. This act is a matter of murūʾah (chivalry/manliness) and embodies the meanings of a man's guardianship and qawāmah (custodianship), and there is reward and recompense in it.