There is plenty of research showing negotiating differences between the sexes (including differences in negotiating based on the gender of the counterparties relative to one another).
Shouldn't the government instead compare discrepancies between initial offers for the same position instead of actual salary negotiated?
If they don't use initial offer, then the business needs to correct for this and every way I can think of to correct destroys the firm's ability to recruit the difficult candidates while still remaining competitive with other firms. The firms that correct for these differences will pay more on average for labor than its competitors and will likely lose out on the capable candidates that know their self worth and demand compensation that reflects that.