Questions about Kim's health flared after he skipped an April 15 commemoration of the 108th birthday of his grandfather, North Korea founder Kim Il Sung. It's North Korea's most important event, and Kim, 36, hadn't missed it since inheriting power from his father in late 2011.
North Korea's state media on Wednesday said Kim sent a message thanking Syria's president for conveying greetings on his grandfather's birthday, but didn't report any other activities, while rival South Korea repeated that no unusual developments had been detected in the North.
Kim has been out of the public eye for extended periods in the past, and North Korea's secretive nature allows few outsiders to assert confidently whether he might be unwell, let alone incapacitated. Still, questions about the North's political future are likely to grow if he fails to attend upcoming public events.
Kim is the third generation of his family to rule North Korea, and a strong personality cult has been built around him, his father and grandfather. The family's mythical "Paektu" bloodline, named after the highest peak on the Korean Peninsula, is said to give only direct family members the right to rule the nation.
That makes Kim's younger sister, senior ruling party official Kim Yo Jong, the most likely candidate to step in if her brother is gravely ill, incapacitated or dies. But some experts say a collective leadership, which could end the family's dynastic rule, could also be possible.
"Among the North's power elite, Kim Yo Jong has the highest chance to inherit power, and I think that possibility is more than 90%," said analyst Cheong Seong-Chang at the private Sejong Institute in South Korea. "North Korea is like a dynasty, and we can view the Paektu descent as royal blood so it's unlikely for anyone to raise any issue over Kim Yo Jong taking power."
Believed to be in her early 30s, Kim Yo Jong is in charge of North Korea's propaganda affairs, and earlier this month was made an alternate member of the powerful Politburo.
She has frequently appeared with her brother at public activities, standing out among elderly male officials. She accompanied Kim Jong Un on his high-stakes summits with President Donald Trump and other world leaders. Her proximity to him during those summits led many outsiders to believe she's essentially North Korea's No. 2 official.
"I think the basic assumption would be that maybe it would be someone in the family" to replace Kim Jong Un, US national security adviser Robert O'Brien told reporters Tuesday. "But again, it's too early to talk about that because we just don't know, you know, what condition Chairman Kim is in and we'll have to see how it plays out."
The fact that North Korea is an extremely patriarchal society has led some to wonder if Kim Yo Jong would only serve as a temporary figurehead and then be replaced by a collective leadership similar to ones established after the deaths of other Communist dictators.
"North Korean politics and the three hereditary power transfers have been male-centered. I wonder whether she can really overcome bloody socialist power struggles and exercise her power," said Nam Sung-wook, a professor at Korea University in South Korea.
A collective leadership would likely be headed by Choe Ryong Hae, North Korea's ceremonial head of state who officially ranks No. 2 in the country's current power hierarchy, Nam said.
But Choe is still not a Kim family member, and that could raise questions about his legitimacy and put North Korea into deeper political chaos, according to other observers.
Other Kim family members who might take over include Kim Pyong Il, the 65-year-old half-brother of Kim Jong Il who reportedly returned home in November after decades in Europe as a diplomat.
Kim Pyong Il's age "could make him a reasonable front man for collective leadership by the State Affairs Commission and regent for the preferred next generation successor," said Leif-Eric Easley, a professor at Ewha University in Seoul. "However, elite power dynamics and danger of instability might make this an unlikely option.