North Korea is still sending letters and gifts to foreign leaders and domestic workers in the name of its leader, Kim Jong-un.

Its news media brims, as usual, with panegyrical propaganda extolling Mr Kim’s leadership.

South Korea reiterates that it has detected “nothing unusual” in the North.

US President Donald Trump has called “incorrect” and “fake” a report that Mr Kim was “in grave danger” after surgery.

All this has done little to stop the rumour mill churning about Mr Kim’s health and the fate of the nuclear state – for the simple reason that North Korea has not reported a public appearance by its leader for two weeks.

Nor has it responded to lurid claims about his health.

The lack of real information from the hermetic country is giving rise to rampant rumour mongering, leaving North Korean experts, foreign officials and intelligence agencies to parse through it all for signs of the truth.

Depending on the news outlet or social media post, Mr Kim, believed to be 36, is recuperating after a minor health issue like a sprained ankle, or he is “in grave danger” after a heart surgery.

Or he has become “brain dead” or is in a “vegetative state” after a heart-valve surgery gone wrong at the hands of a nervous North Korean surgeon or one of the doctors China dispatched to treat him.

Or Mr Kim is grounded with COVID-19.

Where did he get it?

From one of those Chinese doctors.

One rumour circulating in South Korean messaging apps claims that after French doctors could not wake Mr Kim from his “coma,” Kim Pyong-il, a half brother of Mr Kim’s late father, seized power with the help of pro-Chinese elites in Pyongyang, the North’s capital.

It goes on to say that Mr Kim’s powerful sister, Kim Yo-jong, has been detained while Beijing is secretly bargaining with Washington over the future of North Korea and its nuclear weapons.

Seoul has questioned the accuracy of the unconfirmed reports, while the South Korean news media appears to dismiss most of them as rumours spreading through Chinese social media and beyond.

But they cannot be completely ignored, since North Korea is so secretive that the world’s most powerful intelligence agencies have been unable to penetrate Mr Kim’s inner circle.

Mr Kim last appeared publicly on April 11, when he presided over a Politburo meeting.

Speculations about his health began swirling after Mr Kim missed state celebrations for his country’s biggest holiday, the April 15 birthday of his grandfather and founder of North Korea, Kim Il-sung.

Rumours went into overdrive after Daily NK, a Seoul-based website relying on anonymous sources inside the North, reported last Monday that Mr Kim was recovering from heart surgery performed on April 12.

The next day, CNN added to the frenzy, reporting that Washington was monitoring intelligence that Mr Kim was “in grave danger”.

On Saturday, TMZ, a celebrity news website in the US, blared: “N. Korea dictator Kim Jong-un reportedly dead after botched heart surgery”.

More than once, Mr Trump has wished Mr Kim well if he indeed were ill.

“North Korea’s secrecy and our lack of reliable information create a breeding ground for rumours,” said Leif-Eric Easley, a professor of international studies at Ewha Womans University in Seoul.

“But his continued absence would be destabilising as more people in and outside the country wonder if he is incapacitated or dead.”

In recent days, the South Koreans and their allies in Washington have scoured North Korea with the help of spy satellites and other resources for signs of Mr Kim and preparations for missile launches.

Their efforts led them to Wonsan, an east coast town where Mr Kim’s family has a seaside compound complete with yachts, jet skis, a horse track and a private train station.

A train “probably belonging to” Mr Kim has been parked there since at least Tuesday, 38 North, a Washington-based website specialising in North Korea, reported on Saturday, citing commercial satellite imagery.

Wonsan is one of Mr Kim’s favourite sites for missile tests.

A South Korean news report said on Saturday the US had detected preparations for a missile test in Seondeok, farther up the east coast, where North Korea launched missiles in August and again in March in Mr Kim’s presence.

The country has detonated six nuclear bombs in underground tests and claims to have built missiles powerful enough to deliver them to the continental US.

It is also run by a man who was dismissed as a figurehead when he took power in 2011 in his 20s.

Mr Kim has since established firm control, proving brutal enough to execute his uncle, a potential threat to his power, and once calling Mr Trump a “mentally deranged US dotard”.

Donald Trump and Kim Jong-un in the DMZ in 2019. Photo: Getty

In this secretive society, any likely successor to Mr Kim amounts to a guessing game, even for outside analysts who have spent their academic careers parsing the North.

Will it be his only sister, Kim Yo-jong, who has recently expanded her role in his government?

What about Kim Pyong-il, who returned home last year after serving for decades as North Korea’s low-key ambassador to Eastern European countries?

Some predict a collective leadership to be led by Choe Ryong-hae, the No.2 in the government hierarchy.

What if a yet-unknown but ambitious general engineered a putsch?

How would North Koreans who have been trained to worship the Kim family respond?

North Korean leader Kim Jong-un during a visit to Vietnam. Photo: Getty

Officials are careful not to quash the rumours on Mr Kim’s health outright, in part because their past predictions on the North have sometimes proved wrong.

Reporting on North Korea, too, has been strewn with blunders.

Top officials reported to have been executed have often resurfaced.

Some of the defectors, who feed information to the news media, have been accused of, or admitted to, embellishing their accounts.

In 1986, a South Korean newspaper reported a “world scoop” claiming that Mr Kim’s grandfather, then-President Kim Il-sung, died in an armed attack.

A smiling Kim Il-sung resurfaced two days later.

In 2014, Kim Jong-un disappeared for more than a month, prompting rumours he might have been deposed in a coup.

North Korean media later showed him walking with a cane after what South Korean intelligence called an ankle surgery.

In 2015, a North Korean defector claimed that Mr Kim ordered his own aunt to be killed with poison.

But the aunt, Kim Kyong-hui, re-emerged in Pyongyang in January.

The rumours can also turn out to be true.

In 2008, Mr Kim’s father and predecessor, Kim Jong-il, was absent from view for months.

South Korean analysts and the news media speculated, correctly, that he had a stroke.

He died three years later.

Joo Seong-ha, a North Korean defector-turned-journalist for the South Korean newspaper Dong-A Ilbo, said in a Facebook post it was reasonable to believe Mr Kim had health problems.

But he had zero trust in news reports detailing whether and why the North Korean leader faced a grave medical emergency.

Such details about “the health of the Kim family is the secret among secrets,” he said, calling the people who claim to know “novelists”.

The article first appeared in The New York Times.