Whenever Ki-Woong calls Hye-sun via phone, Suk-gyu never even tries to borrow the phone and talk to her directly, even though he presented himself to Ki-Woong as Hye-sun's father. Any parent who is so worried about their child in an accident would surely want to hear their loved one's voice by themselves, but he doesn't. Of course, had Suk-gyu talk to her directly before The Reveal, Hye-sun would immediately tell the imposter as her employer, and the two men would no longer help each other in finding our female protagonist.
