Well this one is easy to answer, because there have been (over the last 6 years) less than say 50 occasions where we
didn't put them to bed together; apart from those 50 occasions, my wife travels a few months out of the year, in which periods I put them to bed. So overall, there is no doubt (no matter how subconsciously biased my 'accounting' might be) that I did the majority of the putting to bed. (to bed putting?)
And well of course there's always the no true scottsman argument - no matter what, one can always put the 'true' equal treatment to question. If you're asking whether I ran a double blind experiment in my home, no I didn't. But we're nit talking about a tiny difference in one observation here. We're talking massive differences in dozens of families (from my observations). And this is for a social context where the ratio if dads and mums and the school gate is roughly 50% (yes I count sometimes), and where the lowest level of education is a bachelor's degree and the median is a PhD. Meaning, we're not talking about representative sample of the population, which you would expect to show the same properties as the population overall; we're talking about a population here that shows high levels of gender equally along many metrics. And despite that, the children show (very) unequal behaviour.