In the UK, people tend to lie to pollsters when they are planning to vote conservative. There have been several elections where the polls wrongly predicted labour would win because of this.
But I was mostly thinking of the poll where men said they wouldn't commit rape, but when the question was re-worded, most admitted they would.
I suspect most progressive men would have a knee-jerk response to a question that used the phrase 'traditional gender roles'. But asking them how often they clean the toilet, or would they want to live with a woman who earns more than they do or expects them to do the housework might have yielded different responses.