Concerns regarding Domain as private input in zk circuit

[Dimo99]

A potential issue has caught my attention: Currently, the domain is defined as a private input within the zk circuit. As a result, this part of the circuit does not undergo verification. I am concerned about the possible implications of this, especially when the sync committee slashing mechanism is implemented. ethereum/consensus-specs#3321
It seems feasible that the sync committee could sign blocks with different domains without running the risk of being slashed
This could, in theory, compromise the security of the circuits.