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Mr. Abe riled Asian neighbors when he was prime minister in 2006 and ‘07 by saying there is no proof Japan’s military had coerced Chinese, Korean and other women into prostitution in military brothels during World War II. He later apologized, but lately he has been suggesting that a landmark 1993 apology for sex slavery may need revising.

Mr. Abe also recently has said he regrets not visiting the Yasukuni Shrine, which honors Japan’s war dead, including executed war criminals, during his time as prime minister.

This issue is important: Former Prime Minister Junichiro Koizumi’s repeated visits to Yasukuni in the early 2000s put relations with China into a deep freeze.

Another front-runner in the Liberal Democratic Party race is Nobuteru Ishihara, son of Tokyo’s stridently nationalistic governor, Shintaro Ishihara.

While the younger Mr. Ishihara is less outspoken than his father, his blood ties would be a major obstacle for Beijing in particular.

“It’s going to be very difficult for him to disassociate himself from his father,” said Jeff Kingston, director of Asian studies at Temple University in Tokyo. “If you do have a nationalist in charge in Japan, they could make things worse. They certainly could throw oil on the fire.”