Open this photo in gallery People watch a news program reporting on North Korean leader Kim Jong-un, at the Seoul Railway Station, in Seoul, South Korea, on April 21, 2020. The South Korean government is looking into unconfirmed reports that Kim is in fragile condition after surgery. The Associated Press

Conflicting reports on the health of North Korea’s Kim Jong-un have created new alarm about the condition of the young, corpulent, nuclear-armed leader with a smoking habit and no obvious successor.

Mr. Kim appears to have skipped an annual birthday observance for his late grandfather last week and did not appear in state media reports on recent missile tests or a sitting of the country’s rubber-stamp parliament.

He last appeared in state media on April 11, and his unusual absence since then has fuelled speculation about his well-being. On Monday, an unnamed U.S. official told CNN that the North Korean dictator is in “grave danger” following surgery. Online new site Daily NK reported last week that Mr. Kim, who is in his late 30s, underwent a cardiovascular procedure related to his obesity and smoking but that his condition had improved.

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The South Korean government, whose intelligence services carefully monitor developments in North Korea, downplayed the reports. The Yonhap News Agency cited a government official who said “there are no unusual signs in North Korea,” including emergency alerts or unexpected movements by the military or political elite. "There is nothing we can confirm with regard to Chairman Kim’s alleged health problem,” said South Korean presidential spokesman Kang Min-seok.

In Beijing Tuesday morning, there was no indication of an increased security presence around the North Korean embassy.

It is difficult to verify information from inside the hermetic North Korean regime, including basic facts about the leadership.

Health concerns appear to be the reason why Mr. Kim vanished from the public eye once before, when he went unseen for roughly 40 days in 2014. He re-emerged with a limp and a cane. And he has a family history of cardiovascular issues: his grandfather, Kim Il-sung, who founded modern North Korea in 1948, died of a heart attack, and all three generations of North Korean leaders have reportedly suffered from diabetes and heart ailments.

Kim Jong-un’s recurrent health issues have raised questions about the stability of leadership in a country that has only known rule by the Kim family, who are revered as gods.

If Mr. Kim were no longer in office, “there would be considerable uncertainty in North Korea’s politics,” said Zhang Liangui, an expert on the country and a scholar at the Party School of the Central Committee of the Communist Party of China in Beijing. It’s not clear how much power the country’s Politburo could wield in an emergency, he said.

“Kim Jong-un is the highest and only leader of the country,” he said. “That is the leadership system.“

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But if unrestrained domestic power has enshrined the Kim family’s rule through three generations, it has also provided an avenue for limitless excess and health risks, through lavish dining and expensive wines and cognacs. Kim Il-sung was said to enjoy rice dishes made with individually sorted grains and apples from trees nourished with sugared water. His son, Kim Jong-il, first developed a love for foreign cigarettes, then domestic replicas, before first attempting to quit smoking in 1982 and finally kicking the habit 17 years later. He even eased up on his hard drinking for health reasons, author Michael Breen wrote in Kim Jong-il: North Korea’s Dear Leader.

The younger Kim, however, has shown no signs of a midlife reckoning with his appetites. In 2016, South Korean intelligence estimated his weight had reached almost 300 pounds, with one lawmaker citing his “habitual binge eating and drinking” as a response to the stress of his position. Doctors who examined footage of him during his flourish of foreign diplomacy in 2018 pointed out his waddling gait and his shortness of breath during even brief walks. He reportedly has a weakness for Emmental cheese and is frequently photographed near ashtrays.

“He has difficulty breathing. He smokes excessively. He looks like a heart attack waiting to happen,” said Anna Fifield, author of The Great Successor: The Divinely Perfect Destiny of Brilliant Comrade Kim Jong Un. “The idea that he has had some cardiac incident or health problem right now rings true.”

She sees his health as a leading risk to his rule.

But if something were to happen to Mr. Kim, who would take his place? It’s not clear whether he has a direct successor. Some reports suggest he had a son in 2010, who would now be 10. “If Kim has a boy, even it’s just a very young child and there has to be a regency, the validity, name and position of the regime will be in the hands of this boy,” said Shi Yinhong, a professor of international relations at Renmin University who believes Mr. Kim has only daughters.

If he doesn’t have a son, “the successor must be someone with Mount Paektu blood,” said Prof. Shi, referring to the mountain that straddles the border with China and has mythical importance for the Kim regime.

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That could be Mr. Kim’s sister, Kim Yo-jong, who earlier this month was reappointed as an alternate member of the country’s Politburo. Ms. Kim was her brother’s envoy to South Korea ahead of the Winter Olympics in 2018 and was a highly visible member of the nuclear negotiation delegations to Singapore and South Korea.

Ms. Kim ”ranks first in a measurement of political power. But whether she is able to take over Kim Jong-un’s power and position, it’s hard to say,” said Prof. Zhang.

It’s also not clear that a woman could assume power in a regime seen as highly chauvinistic.

However, Mr. Kim’s half-brother, Kim Jong-nam, was killed in the Kuala Lumpur airport in February, 2017. He also has an older brother, Kim Jong-chul, about whom little is known – save that he is passionate about guitars and Eric Clapton.

But the millions of members of the Korean Workers’ Party have an incentive to maintain stability and hold on to power, Mr. Breen said. “As long as they’ve got food on their table, nobody cares if the rest starve,” he said. The primary risks are internal dissension, he said, or a leader who lacks the ruthlessness of three generations of Kims.

“If they no longer have the means or the will to kill people, they lose it,” he said.

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With reporting by Alexandra Li

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