SEOUL—North Korea typically puts on a good poker face when a member of its ruling Kim dynasty falls gravely ill. The health of North Korean leaders is among the isolated regime’s most closely guarded secrets—and something foreign intelligence agencies have struggled to keep tabs on.

Kim Jong Un, the third member of his family to run the communist dictatorship, has been out of public view for more than two weeks. He is rumored to be alive, dead, or somewhere in-between.

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