I spotted some other persimmon trees today in the 600 block of West Proctor Street (near Duke Street) in Durham.
Please note these trees are on private property, that of the Duke Physician Assistant program at 800 S. Duke St. Some fruit lay on the sidewalk, but the bulk of it had not fallen far from the tree, so to speak. I recommend asking before foraging. Perhaps you can appeal to the Duke's medical focus by explaining to the powers-that-be that persimmons contain Vitamin A, Vitamin B complex, Vitamin C, lycopene, lutein, beta carotene, folic acid and several important minerals, including copper and potassium. They are also rich in fiber.