There are several evidence that "East Sea" is the proper name.
The "East Sea" is named and called by Koreans for more than 2000 years. It is wrong to call this territory as "Sea of Japan" since more than 70millions of Koreans are using this name.
Since Korea isn't arguing about calling this territory as "Sea of Korea" is because this territory is located closely to several nations and it is not right to name it after an national's name. To call as the "East Sea" seems to be more neutral and efficient.
And since two countries are struggling about this problem. To mark as the "East Sea" and the "Sea of Japan" might be the best idea before an agreement.