FOX has clearly partnered with the Big East, Big Ten and Big 12 (lotta "big" there). They never would have launched this without the three conference partners agreeing to sign on with the terms obligating them to participate. These conferences FOX and AEG can compel their schools to participate, as that is undoubtedly in the contract. (Likely significantly more money and less hassle than the NIT.)

It anybody from other conferences they invite who are not bound. But then again they are not bound by the NIT. ESPN doesn't have any contract with the conferences to attend the NIT or an event of their own. So, in theory even schools from the ACC and SEC could participate in FOX's CBC. To block that, ESPN would have to launch their own rival tournament (not the NCAA's NIT, so competing with themselves) by entering a new contract with participating conferences.

I still think this is a great opportunity for Fox and AEG to create an NIL collective and pay each player $5k for first round, and escalate the payouts each round to a large prize at the end. You will see teams fighting over this invite, hell, may even sway players to join teams in these league for the opportunity.