I see three pieces here, in Texas - ESPN - and the Big 12 conference offices. There is a fourth - in Fox television, but not so much involved.
I see no reason that Texas, currently, is anything but in a position of great strength right now in any negotiation. Not a good position for anybody that wants change. They've got the contract with ESPN. ESPN would gladly negotiate the terms of the LHN contract, I think, but that doesn't mean much - if Texas doesn't want to talk. As for the Big 12 conference offices - they're obviously the piece of the puzzle that can change how Texas feels about talking. That doesn't mean they'll change the position of Texas being in the position of strength in a negotiation, unless they manage to do something, to both motivate Texas to sit down at a table, and put them in position of less strength than they're in now.
The only thing that I think can be done, toboth put Texas into a chair at a negotiation, and into a weak position, is for the other 9 members of the conference to essentially put them on a notice that they're going to get booted from the conference, if the $ numbers are right, such that the conference can continue making what it makes now, with other membership other than Texas - or make more - for it's membership through a conference network and broadcasting deal.
ESPN - I'm sure would have input into how that would work - as well as Fox, and CBS, etc.