You could frame the issue as collective self-image. Also, the argument that people won’t respond strategically because they will have a negligible effect on the overall results rather begs the question of why people bother to fill in the survey at all.
As far as I know, there is no evidence on the issue. Perhaps it could be seen whether relatively objective questions yield different answers by gender. “Has anyone in your department been suspended for harassment”. But there are at least two problems. Participation by gender may differ by department. Second, will gender be reported truthfully? Respondents may, for example, want it to be said, not even women think this is a problem.