I'd imagine there's a pretty strong correlation between belief that race differences are largely innate and belief that gender differences are largely innate.

Assuming there is such a correlation, what kind of thinking would you expect it relies on?

What we really need to see, if this issue is to be approached in a Bayesian manner, is a fully laid-out hypothesis about the causal pathways that go from genes to race to intelligence to IQ, and which of these pathways bypass environmental causal links. If we do have this and are in fact approaching the issue in a Bayesian manner, then success or lack thereof in eliminating gender discrimination can be rigorously treated as evidence.

If someone has not formalized their hypothesis so as to make it testable, then their thinking is anyway of too low a grade to reach reliably correct conclusions based on the evidence, and it does not matter what evidence they look at, they'll end up right or wrong purely by chance.