Kim Jong Un will stop the conflict on the Korean Peninsula if he is able to eliminate his nuclear weapons program and show an effort that he would be able to have and live with a unified Korea. The Korean War and conflict is one that is still being fought over to this very day. It is between two different ideologies with one being a democratic and capitalistic society and the other being a communist stronghold with a leader that received his honor through birthright.
South Korea has long wanted peace but the impression of the international community is that the North Koreans do not want that peace with their neighbor and have always feared that they are vulnerable for attack and invasion by bigger and stronger Western nations. Un is going in the right direction having meetings with the Chinese and a scheduled one with President Donald Trump. The thing is that there will be no progress in the Korean conflict until Kim Jong Un is able to prove that he is about unified peace on the Korean Peninsula and that he is serious about eliminating his nuclear weapons program.