I found this to be really interesting: "The researchers were especially surprised by the way in which the companies adapted to the regulation – by slowing the growth of men’s wages, not increasing women’s."
Did the researchers measure employee departures? If wages stagnated in general, then one would expect that employees who are more desirable/mobile to depart. This would make it seem like wage equality went up, when in reality the people who were previously making more are still doing so — they're just doing it at a different company.
Have not read it closely, but "no statistically significant" need not rule out the scenario that departures drive the whole effect. It's just saying that the error bar from counting departures is large enough to include zero, while the error bars on mean salary are smaller, because everyone has a salary but only a few leave.