Since the controversy over the White House’s new contraception policy broke, it’s been widely assumed that the battle is terrible politics for Obama, because it will cost him among Catholic swing voters.

But some polling from August suggests a majority of Americans supports the White House position — and that the opposition to the provision from the U.S. Conference of Catholic Bishops makes no difference to them. Even a majority of Catholic respondents said the same.

The Democratic polling firm Lake Research did a poll — commissioned by the moderate-to-liberal health care group Herndon Alliance — that tested messages on both sides. The poll was conducted in August, when the new administration rules first broke.

The poll tested the message of those who say Catholic institutions shouldn’t be required to provide birth control coverage, and found a majority see it as unconvincing:

Requiring health insurers to cover contraceptives violates the rights of people who belong to religions that don’t believe in artificial contraception. The Catholic Church morally opposes birth control and Orthodox Jews and some Protestants find birth control objectionable. Forcing religious groups, individuals, health providers, and health plans to perform or pay for a service that they may find morally objectionable is wrong.

Convincing: 42

Not convincing: 52

The wording there is not perfect, but the last sentence comes close to describing the crux of the controversy. Meanwhile, another question directly tested the fact that the U.S. Conference of Catholic Bishops opposes the new rule, and found that for a majority, it didn’t make any difference in their views of the Affordable Care Act:

The United States Conference of Catholic Bishops has sharply criticized the new rule requiring private health plans to cover contraception and birth control with no co-pays or deductibles saying, quote — pregnancy is not a disease, and children are not a health problem — end quote. Does this make you more favorable toward the new health reform law, less favorable, or does it not make a difference?

The responses: 57 percent said it made no difference; versus 18 percent who said it made them look on the law more favorably, and 20 percent who said less.

Among Catholics, 53 percent said the bishops’ position made no difference to their view of the law.

“This poll tells us that the American people believe that women should have access to contraception through normal health insurance,” Bob Crittenden, the executive director of Herndon Alliance, told me. “If the White House is looking at what the American people want, they don’t want to be told what to do by Catholic bishops. Women and men in the United Sates have a strong feeling that this is a good preventive benefit that should be available to all women.”

Such polling needs to be seen in context — it’s an effort by an advocacy group to test messaging in order to persuade its own side (the White House) to stand firm. And the polling was conducted before the controversy’s latest round flared up. But as Sam Stein, who also obtained the same polling, points out, the results are very relevant politically right now.

The White House very well may buckle in this fight. But these numbers do suggest at least the possibility that leading commentators have been far too quick to declare this a certain political loser.