I was part of a research team (class project; never published) that did something like this about a dozen years ago.

We were specifically looking at hiring-related gender bias among different education levels. Our study showed non-statistically significant bias among those without a bachelor's degree, and statistically significant at a moderate correlation level among bachelor's degree holders. The worst were the subjects with graduate degrees; not only was it a strong relationship at a statistically significant level, it was as extreme as our methodology could detect. And it went both ways (e.g., a female identified applicant wouldn't be considered for a mechanic position, but a male identified applicant wouldn't be considered for a nursing position either).

Respondents weren't supposed to put specific information on their responses, but one prominent individual did. Our curiosity got the best of us, so after we input and calculated our metrics, we looked at the identifiable subject's responses. This individual was in charge of EEO and non-discrimination policies at a research 1 institution, and she was the most gender discriminatory individual in the study!!!

The studies I've seen actually demonstrate that women get called for interviews slightly more often than men. (The studies were not in software engineering, so different industries certainly are different).

I've heard more personal experiments (be it by gender or ethnicity) leaning more towards white/male names getting more interviews and jobs than ethnic/female names.

It would be especially interesting to compare by gender the amount of interest, frequency of landing an interview, and rate of success of landing an interview. I'd be very curious to see if biases shift depending on the phase of hiring process.

"The studies were not in software engineering"

Worthless comment then, carry on.

I somehow suspect that the opposite would be true for, say, preschool teachers or nurses.

As far as the Tech industry in the UK is concerned, the polar opposite couldn't be more true.