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Republicans nationally haven’t bolted the former national restaurant chain CEO who has recently risen from obscurity to near the top of national polls with Romney.

A Washington Post-ABC News survey taken after the allegations emerged last Sunday showed Cain and Romney running nearly even atop the field, with most Republicans dismissing the harassment allegations. Seven in 10 Republicans say reports of the allegations don’t matter when it comes to picking a candidate.

But in a sign of the possible danger ahead, the poll found that Cain slipped to third place among those who see the accusations as serious, and Republican women were significantly more likely than men to say the allegations make them less apt to support the businessman.

While the questions apparently haven’t struck a blow against Cain in Iowa, their persistence is giving some GOP caucus-goers pause at a critical time.

A recent poll of likely Iowa caucus-goers sponsored by The Des Moines Register showed Cain narrowly leading in Iowa.

But Cain has a smaller campaign staff in the early states than many of his rivals.

The questions aren’t discouraging Iowa state Rep. Henry Rayhons from siding with Cain — yet.

“He’s got to come clean, or people are going to keep harassing him about it,” said Rayhons. “The longer it hangs out there, the less likely I am to support him.”

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Associated Press writers Jim Davenport in South Carolina, Phil Elliott in Iowa, Laurie Kellman in Washington and Holly Ramer in New Hampshire contributed to this report