I cannot comment much on thermodynamics/biochemistry as I don't know a bit about those, but unfortunetaly it seems like this study has ommited the fact that a high protein diet will always be more efficient in the beginning. They should NOT have taken weightloss that occured in the first week or two (to be on the safe side) into account.

Why? Protein-rich food tends to weight less, doesn't cause as much bloat. Actually a very high fat diet might cause most loss in the beginning.
For example I eat a very high-carb diet,a lot of raw food that is not too calorie dense, it bloats you a lot providing few calories at the same time. If one day I decide to eat only ice cream (keeping ammount of calories the same) then on the next day there will be a sudden big loss. Of course it will be gone once back to eating what I eat before.

So when measuring efficienct of a diet you should exclude anything that happens in the beginning because it's always very very volatile in that stage. Especially when subjects tested ate a very different diet beforehand. You could also go on to talk about sodium intake which affects water retention which translates to changes in weight. That will cause a lot of fluctuation in weight in the beginning too.

I keep a very detailed log of what I eat and my weight with CRON-o-Meter. When I eat and drink same food for an extended period of time (and stay home all the time) my weight loss is almost calorie-perfect, meaning that you could convert the daily calorie deficit to grams and this is how much I would loose on each day. Only once every 4-5 days there would be a slight "bump", but if you look at the 7-day moving average it's virtually a straight line.

I went on vacations once and decided to binge, big time, 6000 calories a day (really!), at the end of the day it was very unplesant, but well the food was worth it, haha. Keep in mind that I weight 110lbs and eat around 1700 on a normal day. The weight of my body has increased probably by around 5-6 kg, 4kg was expected, rest was water retention etc.
So the first few days back on a diet were crazy. One the first day a loss of 1.3kg (and that is after a 2 day fast already!). Next day 0.3kg, then 0.2 kg, 0.15kg, one day of no change and then a sudden increase of 0.6 kg, after that it gruadually stareted to approach the expected rate of loss (0.1kg/day).

Many people, when they begin a diet are very much like me going off a binge. Not as dramatic of course, but similiar. In a choice of different food can yield a big loss that won't be there long term. A lot of fad diets exploit that fact and so do high protein diets.
People often go on those diets, loose even up to 5-10lbs purely because of eating a food that digests and affects water retention in a different way and maybe some other factors that I don't know of (10 would be a lot, but I guess plausible for a very fat individual).
Some get satisfied with that small loss, go back to the old diet and then we hear about the mysterious "yo-yo effect", but in fact nothing strange has happened.
Those that last and continiue the diet will often say that the "starvation mode" has kicked in. No, "starvation mode", the "water retention effect" is gone (simplification).
Some, after hitting the "starvation mode" will even swtich to more calorie dense food increasing their total intake of calories but reducing the sheer volume of food and claim to have beaten the "starvation mode" with more calories.
I even consciously use those tricks to manipulate my weight sometimes (as there it can affect my mood :-) ).

I guess you understand by now. Unfortunately most studies fail to account for that, so whenever I see a study that tries to prove something with a small sample size, conducted over a short time frame I'm very very skeptical.
But then again, once your sample size and time frame get bigger controlling those people gets hard, too hard. You have to make sure they eat the exact ammounts of food they are supposed to eat.
They sometimes give people directions like eat one cup of this, 2 cups of that, CRAZY!
You have to portion the food for them, unless you explain what a "cup" of any given food is and make sure they PERFECTLY understand you will have huge errors even with 100% honest people.
Accounting for bias is very tricky with those studies too, not going to work too good.
Psychological effects actually get stronger and harder to control the longer the time frame is, so large N and a long time frame doesn't solve the problem either.
Physical activity is another thing. Some people burn 200 without noticing.
There's very few people that truly have naturally "fast metabolism", yet a big proportion of people that are thin claim to have that. "I eat whatever I want and stay lean", yea you do eat whatever you want but you don't want to eat much. Or they do sports. Or fidget all the time (which some claim could even burn off 500 in extreme cases).

Do a study with each group having at least 30 people, same gender, similiar weight, similiar height, healthy, close to no physical activity, not a fidgeter. Portion food for them, don't let them do it themselves. Measure everything they put in their mouth, water and multivitamin included. Weight each day at the same time, same clothes, make sure they pee beforehand. Do that for 3 months. Discard first 2 weeks of results. Perhaps the last 2 weeks too (people are more likely to cheat at the end).
That's a study that I might care about, of course probably forgot about 20 other things I'd do.

I think conducting such a study would only be possible if you gathered a large group of dieting/nutrition enthusiasts like me. Too hard otherwise.
I really sometimes wonder how they find people for those larger studies.

Getting my mom to diet was/is a BIG pain even given the fact that she was willing to drop some weight. She was also extremely biased in a lot of things she said even given that she probably knows more about nutrition then the average person as I give her a lot of lectures :-).
Now try that with 100 "normal" people that you probably DO NOT control on a daily basis.
I know for a fact that unless you drug those people, they are BIG enthusiasts or lock in a cage and monitor the bias will be enormous.

I could go on talking for a very long time. The conclusion is pretty simple. Most studies about weight loss are simply useless.
There are many smart people in the field of nutrition I'm sure, unfortunatelly it's very very hard to find these amongst legions of incompetent people.
I mostly see 2 kinds of the incompetent.
One group is those that are simply stupid/lazy/don't care. They do a study to seem smart, probably because they need it to get some kind of a credential. They study something that 100 people already did, change two things, come up with a conclusion that is not too controversial so there's not too many people questioning (when you're findings are interesting everybody is going to point out even a slight flaw in the study, if you say what people want to hear you might get away with a large, fundamental flaw, because you state "the obivious" anyways).
Another group of incompetent people are something I'd call the "ideology-driven", those are scientists that will do anything to prove their thesis. Actually I think those are a little better than the first group of the incompetent. You see, the stupid/lazy ones, they have almost a zero chance of comming up with anything interesting. The ideology driven ones can actually find something interesting.

Ok, well I'm tired writing that long rant on the field of nutrition (but I guess it applies to so many more fields of science).
If you was bored enough to read the whole thing please also keep in mind that I admittedly have no education in that field, or any whatsoever, so everything I wrote is basically based on my personal experience and what I read on the Internet.
But I do not think I have said anything that is far from the truth.

So as to that particular study, their reasoning might be right, but I wouldn't use that data to substantiate it.
They are honest people though as they do explicitly say "one can't predict that the ratios will stay the same over a long term dietbut the calculations show that the possibility of metabolic advantage should not come as a surprise."
And I'm afraid that the advantage would wear off.
Also note that I'm not saying that I know from my experience that it does, because I have never really went long term on diet that's not high carb, I hate low-carb stuff.
I only have tons of experience with short term effects and what I see in most studies are those short term effects that I know a lot about.