Systematic underpayment of women in a free market?

There's still those who would argue that without government intervention, women would systematically be underpaid for doing the same job as men. Those people are in for a big surprise:

But to those persons who believe that women are indeed consistently underpaid, boy do I have a deal for you! Start your own firms and hire only women. If it's true that women are consistently underpaid, you'll be able to hire outstanding employees by paying them more than the relative pittances they currently earn, while you still profit handsomely from employing them.

And that's not all. Being benighted male chauvinists, your competitors will not follow your example; they will stubbornly refuse to offer female employees wages commensurate with these women's productivity. You'll expand your operations by easily hiring highly productive, formerly underpaid workers while your competitors – made stupid by prejudice – will shrivel into bankruptcy as they lose productive employee after productive employee. You'll simultaneously corner your industry's market, earn handsome profits, and raise women's wages. If you're correct that sex discrimination is rampant in today's labor market, you can't lose! So get to work!