Il Mare

There are two distinct time-lines intertwined throughout most of the film.
The story begins with Eun-joo moving out of a house by the sea called "Il Mare". As she is leaving, she leaves a Christmas card in the mailbox, asking the next resident to please forward her mail to her. Sung-hyun, an architectural student, receives her card, but is puzzled, since he is the first resident at "Il Mare" and the card is dated 2 years in the future. After a series of back and forth correspondences, Eun-joo and Sung-hyun realize they are living 2 years apart, Eun-joo in the year 1999 and Sung-hyun in the year 1997. After some testing, Eun-joo and Sung-hyun discover that the mailbox at "Il Mare" is enabling their communication and they can pass objects and living creatures through.
Utilizing the mailbox, Eun-joo asks Sung-hyun to retrieve a tape player she lost two years ago, which he gets for her. After his estranged father, a noted architect, falls ill, Sung-hyun asks Eun-joo to obtain a book about his father, which she does. However, she succumbs to a minor traffic accident and while hospitalized, the book fails to reach Sung-hyun in time before his father's death. After reading the book, he finally accepts his father's love for him and takes up his architectural work once more.
As both Eun-joo and Sung-hyun continue their correspondence, they decide to try a date together, with each person participating in his or her own time. Eun-joo "takes" Sung-hyun to an amusement park, where he follows her instructions on how to have a good time at the park. Sung-hyun "takes" her to a restaurant where she drinks a bottle of wine he left for her two years prior. Despite having a lot of fun on these solo "dates", they decide that they should try to meet in person.
Eun-joo and Sung-hyun plan on meeting in person at a beach where she has always wanted to build a house, two years in Sung-hyun's future. However, when Eun-joo goes to the beach, Sung-hyun doesn't show. She does see a house being built on the beach for an unknown architect's lover. When Eun-joo tells Sung-hyun that he didn't come, he is baffled about why he didn't show up; he doesn't think he would have forgotten such an important date.
At Eun-joo's workplace, she runs into her ex-fiance. They were going to get married, but he moved abroad for work, while she stayed in Korea. Due to the separation, they eventually broke up and he married another person; Eun-joo however still loves him. This meeting was a shock to Eun-joo and in an act of desperation, she asks Sung-hyun to intervene and stop her fiance from leaving two years in the past. After sending the letter to Sung-hyun, Eun-joo suddenly realizes, on that day, she witnessed a car striking a pedestrian and killing him. Eun-joo rushes to Sung-hyun's architectural school and finds out that Sung-hyun was that very pedestrian and the house being built at the beach was designed by Sung-hyun for her. She immediately rushes to the mailbox and sends a letter telling him not to go.
The final scene returns to the beginning of the movie, where Eun-joo is about to place her Christmas card into the mailbox at "Il Mare". A stranger approaches her with a letter in his hand, the letter that Eun-joo sent warning Sung-hyun not to go to the meeting. Sung-hyun did receive her warning letter and never went to intervene that day and was never hit by the car. Eun-joo and Sung-hyun finally meet.

Question: Who was the house being built at the beach designed by?
Answer: Sung-hyun