Found it! It's called Back Home, Here's the author's FF.net account: <a href='http://www.fanfiction.net/u/212642/Warringer' target='_blank' rel='nofollow'>Warringer</a>
And the explanation here is that they're fighting over the irony that they're too much alike. They both think they're the top of the food chain and deserve to be there, and can't stand that there's another who is their equal or even can come close. And the Darker variant, Going Home, has them both outright enslaving other races out of some twisted idea that they're "uplifting" them, though I admit that canonically, that was the Centauri's excuse. Neither Jurai nor Minbar in their canons did that. They did that in Back Home too, but it wasn't as pronounced.
Oh yeah, the Vorlons and the Shadows are waiting in the wings to start another round of their stupid pissing game, and only Tenchi and his chosen few know that and are frantically preparing to survive the storm. Some of the AU to this story have them losing and running to other universes like nBSG (Best option since they can pretty much curbstomp the Cylons easily even in their weakened state, although another alternate on Spacebattles has them gettng attacked by the Colonials with the Colonial military getting handed their heads) Star Wars (Somewhat bad since they have the tech to take on Deathstars and win, but the Empire can still Dogpile them with numbers and even if they win, there are other nasties out there like the Yuuzhan Vong), or worst yet, Warhammer 40000 (right in the middle of a battle between the Tyranids and the Eldar no less, and yes, they know how bad it is since to them, W40K was fictional until they ended up in that universe. Talk about frying pan and the fire).