Rumors are flying fast and furious that the Obama administration, pelted by blows from the Catholic Church and Republicans over its regulation requiring religiously affiliated colleges, hospitals and charities to provide employees with no-co-pay birth control, could be poised to back into a compromise. The impetus for such a concession seems to be the fear that Catholic voters, a group crucial to Obama’s reelection effort, may abandon him over this issue. But what effect would a reversal have on groups that strongly favor the rule as it currently stands?

Our February Religion & Politics Tracking Survey shows that Catholics are divided on this issue, which suggests that Obama may not be committing political suicide among these voters. What’s more, a closer examination of the data reveals, that backtracking on the rule could disaffect two other crucial demographic groups who strongly support the lack of an exemption for religiously affiliated institutions: women and Millennials (age 18-29). So even if the Obama administration manages to pacify the Catholic hierarchy by modifying the rule, a turnaround could stir up resentment among these groups.

Women are, unsurprisingly, more likely than men overall to favor the mandate that requires religiously affiliated hospitals and colleges to provide health care plans that cover contraception, (54% vs. 43%). Generational differences are even more striking: while nearly 6-in-10 (58%) Millennials say that religiously affiliated institutions should provide birth control through their insurance, only one-third of seniors (age 65 and up) agree.

Perhaps most interesting, however, are the differences between younger men and women. Men under the age of 50 are evenly divided on the exemption, but a staggering 61% of women under the age of 50 agree with the current policy, with only one-third (34%) saying that religiously affiliated institutions should be able to opt out of providing no-cost birth control for their employees.

This makes sense, considering that women between the ages of 18 and 49 stand to gain the most from the new rule. According to the Alan Guttmacher Institute, 62% of the 62 million women of reproductive age are currently using a method of contraception. Birth control costs can range from $15 to $50 a month, even for women who have health insurance. The new regulations lift this financial burden from these women’s shoulders, and given their levels of support for the rule as it currently stands, it’s unlikely that they will be happy if this policy is changed.

To be sure, Obama needs support from Catholics to succeed in this November’s election. He’s currently leading Romney among this key demographic (48% for Obama vs. 40% for Romney), which is a slim enough margin that Obama’s advisers concern about alienating this constituency is understandable.

But it would also be a mistake to think that Catholics are the only group that cares about this issue. Young women were a crucial part of the coalition who helped propel Obama to victory in 2008, and the Obama camp will be out to court them again this year. A high-profile compromise might dampen these women’s enthusiasm, to the point where the White House may wish it had taken a step back to look at the whole playing field.