Export changes of a document directly to overleaf web using Mathematica?

Questions
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Overleaf is a nice website which allows collaborative writing and publishing of LATEX documents.

I am currently working on a document. I need to make some modifications to the file there.
To give a simplified example, in the hyperlinked document here I would like to increment each number.

I can accomplish this first by doing the modification on the string obtained from the URLFetch-ed the online file:

text = StringCases[
    URLFetch["https://www.overleaf.com/read/mzshmkjmkxxd"], 
    Longest[ "\\documentclass" ~~ ___ ~~ "\\end{document}"]][[1]];

where the url is the Read Only Link available from the Share button, in the menu;

And then, using the following code snippet:

refinedText = StringCases[text, Shortest["\\section{" ~~ ___ ~~ "}"]][[1]]
StringReplace[refinedText, a : DigitCharacter .. :> ToString[ToExpression[a] + 1]]

I do the incrementing which I can later replace in text and ultimately (this is the task which I would like to automate) replace the document online with text.

So, my question is:
how to upload/export the changes directly to overleaf using Mathematica?

I have tried, without any success, to do that using the Read & Edit Link (also provided through the Share button).
This link may come in handy to those interested in this question.