If you don't want the added cost of litigating a matter in a Korean court, please register your trademark in Korea. No. No. No. Registering in the U.S. and the E.U. is not enough. Your "international filing" only gives you a grace period to file in other nations that have signed onto particular international treaties.
As the China Law Blog notes:
China is a first to file country, which means that, with very few exceptions, whoever files for a particular trademark in a particular category gets it. So if the name of your company is XYZ and you make shoes and you have been manufacturing your shoes in China for the last three years and someone registers the XYZ trademark for shoes, that other company gets the trademark. And then, armed with the trademark, that company has every right to stop your XYZ shoes from leaving China because they violate its trademark.
Replace the word Korea with China and reread.