Can't say I'd be particularly interested in taking part. As a female scientist who knows how many social and cultural barriers there are to female progress in my profession, I'm inclined to take his "the data are in" claim vis-a-vis men liking stuff and women liking people with a pinch of salt. Yes, there are many more men in scientific professions and women in caring professions - but the bare numbers don't tell us anything. They don't tell us why.
We still haven't got a test which can tell us whether it's nature or nurture. Yes, I know his claim is "look at the Nordic countries, they've done more than anyone else to remove barriers to women's participation - yet women still aren't rushing for those jobs" - but as we all know, legislation and workplace culture are two different things. You can have laws on paper saying you've got a right to long maternity leave, your partner gets just as much paternity leave, you have a right to flexible working, you have a right to equal pay. But it's quite possible for all those legislative aspects to be in place and for the culture to still be putting up barriers. Or for those legislative aspects to be in place but "more honoured in the breach than the observance."
(I had to take my last employer to court to get equal pay - the law was there, hence I won, but my employer certainly wasn't going to do it off their own bat. And I was lucky in being in a unionised environment, so there was a union to bear the legal costs and financial risk on our behalf, and in a workplace with enough decent men prepared to step forward that we could get workplace comparators to prove the case. Those are the sort of soft, cultural issues that act along side the law.)
However, Peterson has spent his adult life building up a repertoire of obscure test cases and facts about different legal systems, and is a very polished performer and polemicist - I know that short of putting in inordinate amounts of time picking his writing apart, reference by reference, he'd simply do a Gish gallop. So I'm inclined to view any such webchat as wasted time and effort on my part.
But if other people fancy it... I'm inclined to agree with posters on here and on the site stuff thread - it'd be hard to see how it could be worse than Penny Mordaunt's efforts.