With a three party lease agreement, each party is obligated to pay their fair share. Any break in that lease agreement by any party, the remaining parties can sue for non payment..........I don't think they'd want to do that.
This is not correct. Each party who signs a lease is responsible to the landlord for 100% of the monthly rent. If the tenants have made some agreement among themselves as to a pro-rata sharing of the rent, which is almost certainly the case here, whoever has been forced to pay more than his/her share to the landlord because one or more tenants failed to pay, he/she can try to collect that amount from the party who didn’t pay. But all this is completely detached from the obligations to the landlord and it’s going to take a very expensive law suit to collect from the non-paying co-tenant.