The only way to update the transfer request is to click on the branch that was PRed - so even the original repo owner cannot change the PR by default. It makes sense - for traceability, at least.
So, if you want to finish this work, the best thing you can do is to unlock the source repo, clone on your computer, add the PR repo as remote , checkout PR'ed branch, commit Also push those changes in your own fork and make a new PR, contained in the comments, which he continues and fixes another PR, so the original PR closes when your merges.
In this case, something like:
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source share