To repeat: "If everyone logically arrived at the same logically correct answer"
Which means everyone votes the same, so the statement is correct.
Granted though, I am making the tacit assumption that people are voting without first looking at the poll results. I believe that is in the general spririt of voting, polls, and surveys, and that is how I always vote. To my mind, anybody who has to look to see how other people vote before making up their own mind, is a pretty sorry example of a voter.
However, if everyone did as you suggested and viewed the results before voting, then in general they will be faced with a different set of circumstances and could logically arrive at different answers. But then probabilistic forces would come into play, because you would never know how many other people are voting at that instant. If by chance there was a sudden surge of voters, and by chance they all opted to vote for "in between", then that could make "in between" the most popular answer, and if the poll happened to end just then, then it would spoil everything. That may be extremely unlikely, but it would alway be theoretically possible. The only way to be absolutely certain that it couldn't happen would be to always vote for "most popular".