Their explanation is based on the hypothalamus being out of sync and affecting the hpi axis.
Unfortunately an explanation I've never found satisfying after spending many hours to research it. Attempts to dig deeper into this (science wise) have not led to any answers, since the collection of studies showing overall picture of hormone and receptor expression and activity don't really show anything especially interesting/unexpected that would gel with the 'HPA axis dysfunction' hypothesis. There were some interesting and more sophisticated papers by Broderick and colleagues, but they are still unsatisfying as they don't actually explain the patterns observed. I have discussed my own hypotheses about this on the forum in the past, I'd have to find them again though...
There are stories of recovery all over the place from people doing both Gupta and LP. There have been a number recently in the press who have done LP and I know a number of people who have done it and recovered and I personally know 2 who used Gupta. What other proof do you want? These people were diagnosed with ME/CFS. In my opinion, and it's only my opinion, I think there are multiple causes of ME/CFS and these therapies will help some but not others. Similarly there are some cancers which are treatable some which aren't but those who have had treatment and recovered does that mean they didn't have "real cancer"? The problem is the "experts" don't even know so until they do we have to try what resonates with us in the hope it might work. If it's not for you then so be it but it doesn't mean it might not help someone else.
The thing about anecdotes is we have no way of knowing whether these improvements were spontaneous or not. There are selection biases involved eg 50 people try something, 10 people improve and those 10 people think it works and so go tell everyone else. The other 40 just move on to something else. Meanwhile, it is possible that 20% of people may have spontaneously improved anyway so we have no way of knowing if the treatment actually did anything. This is where controlled trials come in.
I personally don't like to get bogged down in the details (the thresholds of a case definition can be a case of how long is a piece of string), but I do believe in high quality evidence.
This means either
randomised double blinded trials, which unfortunately are only applicable for pharmacological treatments, or randomised trials using
objective evidence as primary outcomes (for nonpharmacological treatments).
For ME and CFS it doesn't have to be complicated - simply using actigraphy to measure activity levels before and at the >6 month followups, and neuropsychological testing (which can reflect any improvements in concentration due to less fatigue).
I am not a fan of studies that are not blinded, that simply use questionnaires as there are many, many uncontrolled biases involved.