Can Two-Way Direct Communication Protocols Be Considered Secure?

We consider attacks on two-way quantum key distribution protocols in which an undetectable eavesdropper copies all messages in the message mode. We show that under the attacks, there is no disturbance in the message mode and that the mutual information between the sender and the receiver is always constant and equal to one. It follows that recent proofs of security for two-way protocols cannot be considered complete since they do not cover the considered attacks.