It’s not 40% fakes – it’s 39% * 61% = 24%. You are also making assumptions about how students lie. I wouldn’t assume all of this 24% falsely claimed that they are 100% URM, and they did so in the same way on 100% of their college applications, rather than prioritizing applications where perceived benefit for misrepresenting race/ethnicity exceeds perceived risk (“reach” where the student believes they are unlikely to be admitted without help, and colleges offering special aid/scholarship for particular races).
I expect it also includes things like respondents saying they are mixed race, which is classified as “2 or more races” in NCES, because they heard they are 1/32 Cherokee. This also doesn’t necessarily mean the college will list them as “2 or more races” when filling out NCES forms. There are numerous potential contributing factors why total may not align with NCES.
What is more clear, is all other surveys I am aware of that asked about misrepresenting race/ethnicity also found a good portion of White/Asian applicants lie about race, so this result doesn’t appear to be a unique fluke. There are also many anecdotal reports, although one cannot estimate percentages well with anecdotes. It’s certainly possible that most persons who say they lie in surveys are actually lying about lying, but that wouldn’t be my default assumption.