I don't think they were measuring "honesty"
The paper's title is "Cognitive control increases honesty in cheaters but cheating in those who are honest" - which is a long-about-way of saying "it takes more effort to act out-of-character than to act-in-character".
As others have mentioned: it does not control for those who are mistaken in believing they have spotted 3 differences - it considers them to be lying.
Given the parameters of the test:
1) the subjects are assured that each pair of images has exactly 3 differences
2) the faster the 3 differences are spotted, the more reward they receive
3) the subjects are not required, in any way, to identify the 3 differences
This is an opportunity optimization problem and measures the willingness of a subject to maximize an opportunity for which there is no negative consequence - aside from the "violation" of "moral" standards. Had the subjects been told that for every incorrectly identified difference a kitten would be drowned, I think the results would have been different.
As the authors write in the paper: Imagine a friend sends you a link to a website where you can illegally stream recently released movies for free. Would you decide to stream the movie which you otherwise would have paid for? If so, how many movies would you stream? On a daily basis we are faced with the conflict between the temptation to violate moral standards to serve our self-interest and to uphold these moral standards. (Without trying to open a can of worms - oops! too late!)
In those 4 sentences they conflate "illegal" with "violate moral standards" - I am certain that any casual student of history should be able to come up with examples where "legal and moral" or "illegal and immoral" are not synonymous. Would an ancestor - 500 years ago - sitting outside a music hall, listening to the music they had not paid for, consider themselves to be acting immorally?
Biting the hand that feeds IT
