given that the one horse is bred for show and had a lazy life, and the pursuers have specialized horses that have years of stamina and endurance training, the outcome should be very clear.
no clue how long the fancy horse will be able to keep going, but given the scenario, the prince willl probably not have thought about this and simploy ride on for how long his horse can. I say have them catch up in the late evening and his horse is no longer able to put up a final chase.