The days of women pushing against gender identity at the plumage and courtesy levels will come to an end with a screeching halt when stacked, integrated systems of LDNLS robotics come into their own.
For two primary reasons:
First, because domestic and working robotics will provide people with the free time to court and interact (something notably lacking in today's multi-job, rushed world.) Since that's actually kind of fun, or at least, I think it is, I am fairly comfortable speculating that courting between real people will become common behavior. Again.
Second, if a fellow can buy a social companion that is as exactly up for sex, cuddling and whatever other interests he has... that whole "I wear pants and cut my hair short and makeup is too much work" thing will evaporate like it never existed if a lady actually wants a flesh-and-blood companion.
Should go the other way at the same time, too — a guy robot would never fart, never forget a birthday, never get blue balls, and always be willing to cuddle, gossip, and admire the yapping miniature dog, despite its mysterious resemblance to 17th century illustrations of the drooling, hideous face of Satan. So there will likely be behavioral changes if, indeed, one wants a flesh-and-blood companion.
It does occur to me that one might not want a flesh and blood companion. That could create an interesting state of affairs. No pun intended.