The claim has been kind of played through a game of telephone (it doesn't help that the original link that every blog from 2014-2015 that is cited everywhere now 404s out). The actual survey discovered 421 vegetarians and 2362 people who tried the diet at one point, hence 2362/(2362+421)= 84.8%. Somehow this has transformed into claims as extreme as "84% of vegans quit in the first year". Regardless, the actual claim is that "of both former and current vegetarians in a survey, 85% of the total were of the former type", and carnists conclude:
Before I was vegan, I was vegetarian for a long time, and one summer I actually ate vegan for most of the latter half of that one summer. Would that mean that between that time and when I went fully vegan, I was a "former vegan"? Or what about people that eat vegan for some significant time of the year like Eastern Orthodox Christians, or people that moved in with a stickler significant other, or people that moved to a more vegan-friendly town and tried it out for a while when it was convenient, but then moved away? I know personally people that are in each of these camps. They would have answered that "yes, I have eaten a vegan/vegetarian diet, but no, I do not currently" at some point in their life. None of these conditions prove any of the downstream conclusions, any more than people who start smoking or relapse somehow prove that being a non-smoker isn't sustainable.
So if you dive into the survey, the 84% figure comes from 2362/(2362+421), which is the ratio of former vegans+vegetarians to current+former vegans+vegetarians. What is interesting is that in a follow-up question when you ask people to check off that they didn't eat particular foods while they were vegetarian, that number of "former" drops down by a factor of two. So people might say "Yeah, I was vegetarian for a while" but then they recalled going out and eating meat on a few occasions. So that doesn't really square up with ethical veganism, and kind of confirms the previous point that these terms are too vague.
Also, how does former veganism even work when it's "I quit because meat was too tasty"? Like, did you suddenly conclude that harming animals is ethical? It's certainly possible, but it's unlikely that a majority of vegans drop off like flies because they are convinced by the tastiness of a steak. Again, vague terms.
The survey itself also acknowledges they polled more towards older white people in the midwest, which is why they found a much smaller fraction of vegans and vegetarians [than the bulk population](https://news.gallup.com/poll/267074/percentage-americans-vegetarian.aspx. Also interesting is that the "former veg" category is a decade older than the "current veg" at the time when they started. This all kind of puts the 85% figure into question, as it may be off by nearly an order of magnitude because they polled demographics that are more likely to be former than current vegetarians, who couldn't even keep their stories straight about being vegetarian.
And it would be great if they didn't lump in former vegetarians and former vegans within the same dataset, but because they didn't release the individual numbers it's difficult to figure out why people quit eating strictly plant-based and if they did, why they started and how that compares to those that didn't quit. Also, this study should include, ideally: paleo, keto, gluten-free, and other diet fads to suss out what the numbers are on quitting fads and how they compare to veganism. Given all these lack of controls and methodological problems in this study, this really doesn't substantiate any of the claims that carnists want it to.
Let's push past that all and just run the same analysis on the former-to-total ratio of just vegans, and we get 896/(198+896) which is 81%, less than that for vegetarians in the same group. This means that vegans are less likely to quit than vegetarians. So if veganism is so hard, why is it that vegans are less likely to quit than vegetarians (or at best within the noise)? If vitamin deficiencies are so prevalent when you eliminate all animal products, and diseases are so common that just require, nay, demand that you eat animal products, why are you no more likely to be a former vegan than a vegetarian? So, even by your own flawed methodology, this doesn't show what you want.