They went to the unusual length of arranging a focus group to test anti-Obama films. Conducted by Frank Luntz, the well-known Republican research analyst, a 30-person focus group looked at three choices: Dinesh D’Souza’s “2016: Obama’s America,” which theorizes that the president’s political beliefs were shaped by the radical “anticolonial” views of his Kenyan father; “The Hope and the Change,” a softer critique of the president that features interviews with disaffected former Obama supporters; and “Dreams From My Real Father,” which posits the implausible theory that the president’s real father is Mr. Davis, and that Mr. Davis indoctrinated him with Marxist views early on.

Focus groups were revolted by “Dreams From My Real Father,” with its conspiracy theory paranoia and dubious evidence. It compares photos of the president and Mr. Davis, noting that they have similar noses and freckles. It also purports to have uncovered nude photos of Mr. Obama’s mother in a bondage magazine.

Mr. Luntz’s clients were not surprised. Their thinking was, “I want to know if it’s as bad as I think it is,” Mr. Luntz said.