Some LDP members want the visit canceled over China's clampdown on Hong Kong and its push to assert claims in the East China Sea.
China is locked in a confrontation with the United States over rights, trade, and security, and Washington may press Tokyo to take sides.
But while Japan shares US concerns about China, the Asian neighbors' economies are deeply intertwined so the next leader must balance security and economic priorities.
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Abe had forged close ties with President Donald Trump so the next leader will be starting afresh no matter who wins the US election.
Pressure on Japan from its main security ally to take on a greater share of the burden of its defense and pay more to host US troops is unlikely to ease.
Japan's ties with US ally South Korea are frigid due to disputes over Koreans forced to work for wartime Japanese firms and "comfort women", a euphemism for those pushed into Japan's military brothels, and the feud has spilled over into security and trade.
Security, Constitution
The government has begun a review of its National Security Strategy following a June decision to scrap a plan to deploy the US-made Aegis Ashore ground-based missile defense system at two sites.
An LDP committee wants the government to consider a strike capability to halt ballistic missiles within enemy territory, a controversial issue for a country that renounced the right to wage war after its World War Two defeat.
The proposal is also likely to anger China and Russia, which could fall within range of any new strike weapons.
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Abe engineered a historic policy shift by reinterpreting the pacifist constitution, passing laws to end a ban on exercising the right of collective self-defense, or defending a friendly country under attack.
But he failed to achieve his goal of revising the post-war, US-drafted charter's pacifist Article 9. Amending that is divisive and the new leader must decide if it's a priority.
Public opinion, general election