Some of this discussion goes to one of the central issues, which is that it is possible to test these things if you organize in advance, and the Americans did this under a very similar circumstance with the American community survey, which is mandatory and is in many ways their analogue to our long-form census.
Somewhat counterintuitively, because had you asked me to guess, I would have agreed with you that what is called “item non-response” would be higher under a voluntary circumstance. They found that not to be the case. “Item non-response” was at about the same level whether it was mandatory or voluntary.
On the other hand, it was also that multi-year and I suppose extremely expensive and detailed research that the U.S. Census Bureau did that showed that non-response bias was a very significant problem. They concluded that, no matter how much effort they put into a voluntary census, it couldn't meet their quality standards because of the non-response bias.
In some ways, to me the lesson is that what is really important when you're about to engage in a major change like this is to do the necessary testing and research in advance before you make your decisions.