Bad news for the GOP on the religious liberty vs. contraception debate: Americans aren’t buying what you are selling.

A new Public Religion Research Institute poll released yesterday—which was done in partnership with Religion News Service—found that a majority (56 percent) of Americans do not believe that the right of religious liberty is being threatened in America today. Even worse for the right: A majority of Catholics (57 percent) and independents (58 percent) do not view Obama’s contraception mandate for religiously affiliated institutions as an infringement on religious liberty.

They are also losing on the key argument of the GOP, led by superstar Senator Marco Rubio: that religiously affiliated institutions should have the same protections as churches. PRRI’s chief executive, Dr. Robert P. Jones, told me, “Americans do believe that churches are special. There is no demographic that thinks that churches should be required to [provide contraception]. But a majority of Americans and Catholics continue to think religiously affiliated institutions should be required to cover birth control with no cost.”

The raging public debate and drumbeat from Republicans that Obama is infringing on religious liberty has been largely ignored. Says Jones, “There hasn’t been much of a change since our last poll. The numbers have been consistent. The bigger picture is most Americans and Catholics just aren’t connecting dots between the threat to religious liberty and the contraception debate.”

The GOP seems to be appreciating the lack of traction their arguments are getting and that they may actually be backfiring by alienating independent voters, who make up one-third of the electorate. On the Hill, House Republicans have backed off their efforts to overturn the rule mandating that health-insurance companies provide women working for religiously affiliated institutions with free contraception.

Democratic pollster Celinda Lake told me, “[Republicans] are losing because it’s energizing unmarried and single women. They are losing because their argument is being solidly rejected by independent women. And they are losing because women can’t believe in these times we are continuing to discuss this issue.”

PRRI’s Jones points out that Americans care deeply about religious liberty. It’s just that they don’t see the contraception rule as an infringement of it. Of the people who cited religious liberty as being under attack in America, PRRI/RNS asked people to tell them in their own words where they thought it was being threatened. Contraception placed at the bottom, with only 6 percent of people citing it as a top religious liberty issue. The most frequently mentioned reasons are the removal or God and religion from the public square (23 percent), government interference in religion (20 percent), and hostility toward Christians or religion (10 percent).

Despite the adversarial approach of their critics, the Obama administration continues to work to address the concerns that have been raised. The Department of Health and Human Services announced Friday the release of a new advanced rule that lays out more of the specifics of President Obama’s accommodation announcement for ensuring that religious organizations’ funds will not go toward contraceptive coverage and that insurers will bear the burden of this requirement. The administration addresses concerns over self-insured plans at religious universities and colleges and clarifies that the government is not creating a precedent over what is or is not a religious institution. Critics have raised both issues in their protest of the rule. The public will have a 90-day comment period to respond to the new information prior to the rule going into effect.

Another concern of critics—what do to with Christian insurance companies—has not been resolved, but the administration has asked for public comments providing options for rectifying this situation (PDF). It has said repeatedly it is committed to finding a solution that respects religious liberty.

Jones says that while Americans care about protecting religious rights, “Americans don’t think religious liberty is a right that trumps everything. They are weighing religious liberty with other kinds of considerations, such as obligations of employers, the needs of women employees, and the type of organization.”