You would need a large group of people with regular and often made metabolic (extended, incl. thyroid, cortisol, liver hormones; urine NH to check protein catabolism...) panels during a strict diet. Probably also direct BMR measurements. I am not aware of such studies and not for lack of effort to find them.

I only have my own metabolic panels taken at my expense weekly during a vegan diet (vetted by a dietician) attempt. A failed one to boot. From correct baseline to borderline hypothyroidism in a matter of weeks. Quick rebound when the attempt was stopped.

The study would have to be careful to exclude people with known preexisting conditions that cause obesity and only take in "idiopathic" subjects.

The most interesting related study I found is the Right Track long-term study of adolescent health related behaviour and total factors. It is currently in progress. Hopefully it won't disappear.

I have a few small ones to show:

https://www.ncbi.nlm.nih.gov/pubmed/8772559

https://www.ncbi.nlm.nih.gov/pubmed/2185664