I am curious to know how the ring and the inserts are being supported in the simulation while these interferences are being resolved.
Since frictionless contact is present in each contact, how are you supporting the axial position of each of the four inserts after the contact is established? If you have no axial support, that could cause a failure to converge.
For the ring and the inserts, I recommend a Remote Displacement. Before contact is established, the insert can have all six DOF fixed, and after, make free all DOF except axial and rotation about the axis.
One idea is to allow all four contacts to be resolved on the first time step. To get the sequential states of stress, you can deactivate one contact on step 2 and another on step 3 and the third on step 4. Since the contact is frictionless, the state of stress should be identical to activating them one at a time.