I think science works by proposing models and checking if data falsifies them. A popular model of gender in labour markets is that women and men are on average the same, but that they face discriminatory institutions and employers which lead to them having different outcomes.
Lots of data, on its face, seems to contradict this model: papers suggest that a man and a woman with the same job have the same pay; women earn (very slightly) more than men aged 20-40; women get promoted more aggressively than men if they don't leave the workforce; that women don't seem to rate discrimination the most important thing blocking their progress themselves.
But the model can be modified: let's say men and women are cultured by society into doing different jobs, that qualified women aren't promoted equally; that women earn more but only because they do more education; that so many women are pressured out that the ones remaining are actually much better than the remaining men so they are still under-promoted; that women don't want to speak out or that discrimination is hidden so they don't notice it.
But I wonder whether the model can contort itself enough to fit around these stylised facts:
- Srivastava & Sherman (2015) find that employees working under female managers do not experience smaller gender wage gaps.
- Charles (2011) finds that more gender egalitarian countries often have more sex-segregated occupations.
- Cech (2013) finds that both men and women enter fields that fit with their self-expressed characteristics, net of their beliefs about gender (i.e. even after taking those beliefs into account and controlling for them).
- Charles & Bradley (2009) found that people's curricular field choices are more sex-typed in more developed, egalitarian countries.
- Ceci, Ginther, Kahn & Williams (2014) found (among a huge number of other things) that the higher the average quantitative GRE score in a field, the fewer women were awarded PhDs in it.
- A 2015 ILO report found that women rate direct firm discrimination among the least important barriers in the way of their success in management.
- Gagliarducci & Paserman (2014) found that female-managed firms did worse on practically every measure, but that this was just because they managed different kinds of firms to men.
- Lippa, Collaer & Peters (2010) found that the more gender equal a country, the bigger the gap in mental rotational tasks (a key measure of visuospatial skills) between girls and boys.
- McGee, McGee & Pan (2014) found that male-female differences in competitiveness explain more of the gender wage gap in the NLSY97 (national longitudinal survey of youth that started in 1997) than in the NLSY79.
- Bertrand, Black, Jensen & Lleras-Muney (2014) found that the Norwegian law requiring all publicly traded firms' boards to be 40% or more female had no knock-on effects for women in general.
- Finally Schmitt, Realo, Voracek & Alli (2008) found that men and women differ across the world in personality, but most in developed countries where women have more rights and freedoms.
I bet you can make a model that fits, but that doesn't make it plausible.