You're expressing interest in a potential study, which is nice, but my point is that the study you're proposing is infeasible because it hinges on a non-scientific concept.
When I say that it's epistemologically difficult to find evidence for gender/sex orthogonality, I don't mean that it's a difficult study. I mean that it's difficult (if not impossible) to empirically observe gender (in the sense of "gender theory"), thus precluding any meaningful study from taking place.
In other words: we could do the measurements you're propose, but then we have no way of knowing what it is we're actually measuring. Gender, as defined by gender theory, is inscrutable.
So again, what are you interested in discovering? What do you see as the possible outcomes of your proposed study and what conclusions would you draw from those results? Maybe there's a better way of answering those questions, and maybe these questions are better suited for a different academic field (e.g. sociology).