There will be no movement from any GOR conference prior to 2024.
There is no shake up on the horizon that will help UConn. UConn has to help itself.
I disagree with your statement. It all comes down to money. How much money are you willing to forgo in the short term and how much you are willing to pay to move to another conference. Let's look at GORs (and not focus on exit penalties).
Let's assume that all GORs are rock solid in the Big 12 and the ACC. If you are a Big 12 team, you currently hold your tier 3 rights. If you are an ACC team, you do not hold your tier 3 rights. Notre Dame holds their football media rights.
Thus, if you are a Big 12 team, you can leave the conference by giving up the revenues for basically your home football and basketball tier 1 and tier 2 games. This amounts to less than half of your annual conference payout.
Let's say Oklahoma is getting a conference payout (including bowls, NCAA tourney credits, etc.) of $30 million per year from the Big 12 and they are making $6 million per year from their tier 3 rights. The SEC or Big 10 could offer them $25 million per year in conference payouts including tier 3 rights (lower than full members) until their GOR expires from the Big 12 as the new conference would not have the rights to most of Oklahoma's home football and basketball games. Would Oklahoma take the short term pay cut? Most likely to find a LT home for their school.
For ACC schools, the math is more difficult as they don't own their tier 3 rights. Still, it all comes down to money. Would a school like FSU accept a lower payout in the short term to move to what they consider a more permanent conference? Sure. If FSU asked alums/boosters/sponsors to raise $50 million+ to make up a loss of revenue to move to the SEC, do you think they could raise it? Absolutely.
Finally, Notre Dame is a wild card. They could decide to join the ACC for all sports. Or, they could exit the ACC with minimal pain as they receive a reduced conference payout and own their football rights. Sure, Notre Dame probably has some contractual responsibilities to the ACC, but they could gradually move to a new conference ahead of time.
By the way, even if no school challenges a GORs, announced conference movement will start at least 3 years before GORs expire. By 2020 or 2021, conferences will be looking to add years to their media contracts to keep their conference together. If a conference extends their media rights early, it will signal that their conference is strong for the long term. No school is going to wait until the actual date GORs expire to announce they are moving to a new conference in the future.