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Still, other evangelical organizers were skeptical of Mr. Romney’s efforts and said that any Election Day surge will be despite, not because of, the campaign. They say his tightly knit Boston headquarters hasn’t lifted a finger to win over evangelicals in Ohio or anywhere else.

Historian Doug Wead, an influential evangelical in the Reagan White House, said it’s not the born-again Christians’ attitude toward a Mormon candidate “but this candidate’s attitude toward them” that poses a problem.

Evangelicals resent the fact that many of the top people allied with Mr. Romney are hostile to evangelicals in politics, Mr. Wead said.

“If Romney wins, it will be because of the economy. If he loses, it will not be because he is a Mormon and because evangelicals hesitated to vote for him,” Mr. Wead said. “It will be because he made no effort whatsoever to reach out to them. It was one of the worst efforts ever. It’s not that the Romney campaign made mistakes like John McCain’s 2008 missteps, but that Romney and his people did nothing to reach out to evangelicals.”

He said some evangelicals will stay home on Election Day.

“Evangelicals would have easily fallen into line against Obama if Romney had only humbled himself and asked for their vote,” Mr. Wead said. “He didn’t. So some will indeed sit it out.”

Mr. Blackwell said that leaving the outreach to local organizers was the right call for the Romney campaign.

“His people in Boston have not tried to tell us what to do in our own state,” Mr. Blackwell said. “They have gotten out of the way and let us organize evangelicals and Catholics in Ohio our way.”

Mr. Wead said some evangelicals will find it difficult to forgive Mr. Romney and his aides for what Mr. Wead called the “GOP gang rape of” Rep. W. Todd Akin of Missouri, an evangelical and tea party favorite who in his U.S. Senate campaign said women’s bodies have ways of rejecting pregnancies that result from “legitimate rape.”

Mr. Romney criticized Mr. Akin’s remarks, but Romney allies went further, Mr. Wead said, noting that Republican strategist Karl Rove told GOP insiders, “We should sink Todd Akin. If he’s found mysteriously murdered, don’t look for my whereabouts.”

He said that “both Romney and Rove [who is associated with the pro-Romney super PAC Crossroads America] have withheld GOP money from Akin.”

Mr. Blackwell said Mr. Obama’s “attacks on religious liberty,” such as his health care mandate requiring religious schools and charities to pay for contraception they morally oppose, has antagonized Catholic bishops and their congregants.

“That has also put what we used to call the ‘Reagan Democrats’ into play,” Mr. Blackwell said, adding that this issue should offset the lack of the gay marriage ballot question that helped drive evangelical and conservative Catholic turnout in 2004.