It doesn't, couldn't find any research confirming it.
I also tried to search a lil bit about androgen sensitivity increasing with age ,but couldn't find some clear evidence. Going by the hypothesis that AR sensitivity increases with age ,one as he ages, should also get extra added benefits from androgens(besides having his hair fall off) like increased muscle mass,libido ,etc which doesn't seem to be the case. I also searched about 5a reductase expression increasing with age ,part of the hypothesis that as man ages, his testosterone lowers , the body converts more of it to DHT to satisfy bodily functions, but haven't found neither in this case something that could confirm it.