I was surprised to learn that both for the Moving Sofa Problem and Packing 11 Squares solutions have been proposed, but in either case the optimality of the proposed solution is, as of yet, only conjectured. At first glance, these problems appear "innocent" to me, but presumably deceptively so. (Rest assured: I don't have clue on how to approach either of them.)
What is the nature of the presumed hurdle that needs to be taken here to prove optimality? Is the issue some kind of computational complexity that needs to be overcome, does it require some technique that as of yet is unknown, does the optimality hinge on some other conjectures, ...? Or is this something we don't and cannot know until it actually has been done?
I realize that these are two different problems, but, I thought, perhaps unwisely, to put them in one question. Perhaps they share a difficulty that is shared by many such problems and still is rather specific to such problems.
I include an animated image by Claudio Rocchini of (unfortunately) a non-optimal solution to the first problem. Just because it looks good.