A visit by a North Korean delegation to a Vietnamese city on a popular train route from China has sparked fevered speculation that Kim Jong Un could travel by rail - or road - for his summit with US President Donald Trump.
Speaking to reporters at a White House event on Tuesday, Trump said that if North Korea did not conduct any nuclear tests, then he was not under pressure to achieve one of his leading foreign policy goals - convincing the country to abandon nuclear weapons.
"We would not oppose diplomatic normalization if all the abduction victims are immediately returned altogether", 80-year-old Shigeo Iizuka, who heads a group of the families, said in line with the message they wrote. I think that North Korea and Chairman Kim have some very positive things in mind.
Shortly after Biegun's negotiations with Kim Hyok-chol at that time, Trump announced that Hanoi would host his talks with the North's leader.
The ultra-secretive North Korean leader has not publicly confirmed next week's meeting with Mr Trump in Hanoi, which will be his first visit to Vietnam.
Specifically, discussions on declaring an end to the 1950-53 Korean War could have been on the table, with Biegun last month saying Trump was "ready to end this war".
Moon and Kim have held multiple summits in the past year and agreed to cooperate on everything from reducing border tensions to launching a joint bid for the 2032 Olympics.
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Moon told Trump his country was willing to open economic engagement with North Korea if it will hasten Pyongyang's denuclearization, Moon's office said.
Earlier on Tuesday, Trump spoke with South Korean President Moon Jae-in about the summit.
"We're determined to take up that role if President Trump asks, if that's the way to lessen the USA burden", Moon said.
Moon's praise echoes Trump's own assessment of his efforts, given in a Rose Garden news conference on Friday in which he made the case that he deserves to be awarded the Nobel Peace Prize for his work on North Korea and Syria, but complained he probably would never get the honor.
"He may surprise some but he won't surprise me, because I have gotten to know him & fully understand how capable he is", said Trump.
Wong was a member of the United States delegation, led by Biegun, to Pyongyang early this month.