In 2015, two Obama cabinet secretaries encouraged schools to try a new way of handling free lunch: give it to everyone, no family paperwork required.

The hope was that the expanded program would “both improve child nutrition and reduce administrative burdens,” wrote Arne Duncan and Tom Vilsack. Others believed the lunch and breakfast for all would reduce any stigma associated with the free and reduced-price lunch program.

Now, a new study suggests the program succeeded on one dimension, making students in at least one state slightly healthier in the process. Participating in the program increased the share of a school’s students in the healthy weight range, according to the Georgia State University researchers, Will Davis and Tareena Musaddiq.

It’s an encouraging finding for the initiative — perhaps the first time its health effects been carefully studied — and aligns with other research showing that students benefit academically from access to food stamps and school meals. Yet a substantial number of eligible schools still don’t participate in the initiative, raising questions about whether lots of students are missing out on its benefits.

“If we’re able to find these benefits for the program on health, it may be … important to reconsider participation for schools that are eligible and are not participating,” said Davis.

The paper, which has not been formally peer-reviewed, examines the federal “community eligibility” program as it played out in Georgia schools. It finds that the program did mean more students in eligible schools received free meals, as expected.

When they looked at its affect on students’ health, they found that it increased the share of students in a healthy weight range by 1 percentage point and led to a small drop in average body mass index. The effects were most pronounced in elementary and middle schools.

The health effects are something of surprise, since some past research has found that the federal free- and reduced-price meal program may actually contribute to obesity.

The authors of the latest study hypothesize that improvements to health and quality of school lunches may have played a part. The same law that created the “community eligibility” program also changed nutritional standards for school meals.

“Prior to the … revised standards, meals served in school may have been of lower quality relative to meals brought from home,” write Davis and Musaddiq. “In light of these nutrition standard changes, it is generally important that we revisit the relationship between school meals and child health.”

The Trump administration recently halted other rules put in place during the Obama administration designed to further cut sodium and increase whole grains in school lunches.

Considering its encouraging findings, the paper concludes by noting that a large number of eligible schools in the study — 43 percent — did not participate in the free-lunch-for-all program.

Nationally, nearly half of qualifying schools didn’t take up the program in the 2016–17 school year, though that number that has fallen over time.

One explanation is likely that some eligible schools are not fully reimbursed for the costs of meals, meaning districts could take a financial hit from participating.

Another possible explanation is a bit ironic. The program was intended to ensure students are not stigmatized for getting free meals, but the researchers suggest some schools may be responding to stigma around the program itself.

If that’s true, the authors write, “the choice may come at the expense of forgone improvements to the health of their students.”