“We found no evidence of social-desirability bias on the question about same-sex marriage and have no evidence that Americans are systematically lying to researchers about their true opinions,” Jones says. “This point is further supported by the convergence of findings across hundreds of surveys conducted by dozens of different polling organizations, which all point to the same thing: a real sea change in American attitudes about same-sex marriage.”

Jones is not the only religion researcher who seems skeptical of the idea that a significant number of people are lying to pollsters. Ed Stetzer of LifeWay Research notes that while cultural pressure can produce a “halo effect” that might inflate numbers in polls, “acceptance of same-sex marriage is the trend either way.” David Kinnaman of the Barna Group also acknowledges that the possibility of bias, but he also concludes “almost all the polling points to the fact that attitudes on same-sex marriage and LGBT issues are changing.”

I can only find a single study that suggests social desirability may be at work on this matter, and that study estimates the influence at about 5 to 7 percent. So conservatives’ best-case scenario would only reduce the shift in American attitudes by a nominal amount.