I don't think there's alot of evidence to support the "different wiring" from birth, but plenty to support the effect of conditioning (i.e. imprinting gender stereotypes) on children up to say age 4 when the brain is still growing and forming. Different conditioning - for example not to observe dos and don'ts based on sex, can (from my understanding) contribute to different kinds of brain development.
In any case, there is even more solid evidence to show variations in brain structure, other organs, biochemical makeup, etc. between for example, people of different ethnic backgrounds, people at different age stages in their life, people with illnesses or disabilities and those without. To be consistent, if one argues that men and women should be regarded differently and have different aptitudes because their brains are "wired differently," they'd have to insist in the same for people of different ethnic origins, ages and those with disabilities.
Different doesn't have to mean inferior or superior but our society tends to assign those attributes along lines of difference, like gender but also ethnicity and ability/disability. Imho, there's nothing innate about it.
But, the social conditioning can be very powerful. Gender roles are perpetuated and reinforced through the social, economic and political traditions and messages all around us. They are so pervasive that most of us see these as "normal" or "just how things are." Thinking outside that box can be scary, risky, so a bit like flat-earthers, scientific evidence will do little to shake their belief. To be honest, there's little point in trying to argue with someone who feels such a strong need to cling to beliefs "just because they are" without even being open to questioning. No amount of evidence will make a difference.
Dragging them out of their comfort zones probably won't work, but nudging them might. Little, every day examples that disprove the "rule," and relevant examples of the impact of injustice dropped into conversations can sometimes chip away, oh and seeing that you didn't fall off the edge of the earth (so to speak) when you departed from and started to question the "norm" can also be a nudge.
I really do think sometimes folks need to believe that there is a plausible explanation, some grand reason why there is injustice, either to justify their own privilege OR avoid facing the thought that they are treated crappily for no good reason other than it benefits those who are more privileged for them to be so.