Seems he is balding the way his brother and father did.
He may just not care. And from what I have seen, the royals don't go in much for anything that is not ultra-conservative.
The Queen never colored her hair (however Princess Margaret did).
Charles didn't do anything about his MPB, nor did his dad, Prince Philip.
Probably not considered proper. I assume Harry's beard is the limit of scandal he can handle (given his appearance).