I deleted my previous post on this topic. It contained much that was wrong. If you received it by email, please ignore it. I should by now know better than to post anything I write late in the evening.
Yuan Yu and I did a fair amount of thinking about the interaction of symmetry reduction and property checking. Unfortunately, none of what we deduced was written down and I remember little of it. I intend to think and then write about this, and to check what I write with others before posting it.
Leslie