I don't see how you could possibly know that without knowing where the error in my reasoning is unless you already know with high confidence that in the correct solution the options are either nowhere close to being balanced or identical in every way anyone with consistent preferences could possibly care about.

That's the case. Updateless analysis is pretty straightforward, see shokwave's comment. Solving the thought experiment is not the question posed by the post, just an exercise.

(Although seeing the difficulty many readers had with interpreting the intended setup of the experiment, including a solution might have prevented such misunderstanding. Anyway, I think the description of the thought experiment is sufficiently debugged now, thanks to feedback in the comments.)