Only one country I know of that requires women to cover. And they don't require burkas or chadri's, actually. They just require a woman to be covered from neck to feet. I've been to Mecca (Saudi Arabia) and I didn't wear a burka, just a loose-ish long sleeve shirt and long skirt. My waist, (ah, when I had one!) was evident. I covered my hair with a simple scarf. I was not harrassed in any way.
Some areas have customs requiring it, and in places where extremists are in control (terrorists) it is required for safety, but I am not aware of anywhere where a 'burka" is required by any official law. To say that it is required by custom is sometimes true-but there is usually a bigger explanation than simplistic comments such as Princes' .
BTW, it is just as much "required" by custom that men wear long tunics or gowns, even (kameez or thobes) to protect THEIR modesty. Folks usually miss that when they start going off on women's customary clothing. When a man wears western clothes, they go down a few notches in respect among many Muslims. Some try, like their wives, to reach a compromise acceptable to modesty and the western standards for business wear.
Prince's clothes would be considered pretty raunchy in that part of the world.