If productivity is kept at the same level, and people work a day less, you wouldn't produce the same amount.
That's a huge statement there. Do you have any evidence that this is the case for everyone at every company no matter what?
Of course not, you are asking for the impossible.
But we have evidences that 4 day weeks boost productivity at least in some settings. If you are interested in the topic, look up "4 day week productivity" on a web search engine, or "4 day week" / "4 day workweek" on HN [1,2] if you want previous discussions about this. The topic is not new here. You'll find many posts where companies think it works.
[1] https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...
[2] https://hn.algolia.com/?dateRange=all&page=0&prefix=true&que...
You are not really going to get your average companies to participate in those kind of studies anyway. Those companies want warm bodies and it has nothing to do with empirical evidence