Here's some of what I've found in searching for it:
https://www.alternatehistory.com/discussion/showthread.php?t=166479
https://www.alternatehistory.com/discussion/showthread.php?t=94004
https://www.alternatehistory.com/discussion/showthread.php?t=52917
https://www.alternatehistory.com/discussion/showthread.php?t=94669
https://www.alternatehistory.com/discussion/showthread.php?t=151924
Search for "centralized" "HRE" for more results of the same sort - never tried "united", but it amounts to the same thing for most of the empire's history.
There's also a couple threads recently - one started by me - on "Okay, so what would be necessary for this to work?" in the Hohenstaufen (1153-1250, more or less) period, but I'd start with those above.
I'm not counting timelines involving a successfully united/centralized HRE, but we have several of those here.
I recommend Faeelin's Prince of Peace, myself.
Are you looking for information the HRE uniting with any particular POD, or just "is this at all possible ever, or is it doomed to be what it was OTL?"?