Where is the proof of a generalized integral for integer heights?

Under "other integrals" it gives the integral of any integer height but provides no proof or citation. I remember seeing a paper with similar formula some time ago, but at the end it said something about its conjecture being unproven. Is there any reference that proves this integral formula?
Are we certain that the form posted on Wikipedia is correctly derived?  Without the proof there may be no way of knowing it's correctness, We could take a look at the first few cases of the integral in order to inspect the forms, and then conjecture about a generalization from there, have you derived the first few forms of the integral? 

\( \int\limits x^x dx = I_1 \)

\( \int\limits x^{x^x} dx = I _2 \) 

\( \dots \) 

That may be a starting point for a proof.  Also, I may be misinterpreting something on the wikipedia page that you posted, but shouldn't there be constants of integration in all of these antiderivatives as they do not have limits? 
This may also be of some use: https://en.wikipedia.org/wiki/Puiseux_series

Thanks for the thought provoking question! 

I am not sure it is correct at all, that is why I am asking for the proof. I think it's likely someone found that old paper with the unproven conjecture and stuck it on wikipedia without reading it carefully.

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