Take a look at Scandinavia. They have the moved more towards gender equality than any country in the world. The personality differences between genders have increased, and the proportion of women choosing careers in STEM fields have decreased.
So how much of what they can and can't do is innate? I don't think we can say with any certainty.
We most certainly can, and there's a ton of data to back it. The more a culture allows free choice, the larger the differences become for career choices between men and women.
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