TOKYO—Japanese COVID-19 vaccine minister Taro Kono’s chances of becoming the next leader of the ruling party, and subsequently prime minister, were boosted on Tuesday when a rival’s party faction splintered. Japan’s Liberal Democratic Party (LDP) will hold a leadership election on Sept. 29, after Prime Minister Yoshihide Suga announced last Friday he was stepping down. The winner of the vote is all but assured to be Japan’s next prime minister. So far only former foreign minister Fumio Kishida has announced his candidacy, but Kono and former internal minister Sanae Takaichi have expressed an ambition to run for the post in recent weeks. Kono is considering holding a news conference by the end of the week to announce his candidacy, domestic media said. Following a meeting of former Japanese defense minister Shigeru Ishiba’s LDP party faction on Tuesday, lawmaker Mamoru Fukuyama told reporters members were divided about whether to support Ishiba …