It seems self-evident that eliminating tuition at public colleges for most families, as first Bernie Sanders and now Hillary Clinton has proposed, would increase access to higher education for low-income and minority students. It would reverse one of the key trends limiting opportunity for lower- and middle-income young people: a sustained shift of the cost of public higher education from taxpayers to students and their families.

But without the proper safeguards, such a program might still, paradoxically, narrow access. That’s because tuition-free public college could compound the increasing stratification of post-secondary education into a two-tier system that slots most low-income and minority students into the least selective institutions with the fewest resources and reserves admission to elite campuses mostly for kids from the upper middle-class and beyond.

If tuition is eliminated at public universities for families with income up to $125,000, as Clinton has proposed, more upper- middle-class students who now attend private schools may decide that Austin, Ann Arbor, or Berkeley are better bargains—and intensify competition for the limited slots available there. “What this will do is create a lot of people competing for spaces at public institutions and it will have a bumping effect,” said Anthony Carnevale, director of the Georgetown University Center on Education and the Workforce. “For minorities and low-income students it will push them down the selectivity queue, toward open admission and two-year colleges.”

Landmark research from Carnevale’s center already demonstrates how the post-secondary landscape has evolved into a “separate & unequal” system, as the group calls it. Although far more Hispanic and African American students attend post-secondary schools than two decades ago, most of them have been channeled into the least selective two- and four-year public schools which have the fewest resources to invest in students—and which produce the weakest outcomes in completion or career earnings. Meanwhile, the nation’s 468 most competitive schools remain about four-fifths white, virtually unchanged from 20 years ago, and heavily tilted toward more affluent families. “Right now we have … probably the most stratified college system in the world,” says Margaret Cahalan, vice-president for research at the Pell Institute, which studies higher-education trends.

That powerful sorting shapes not only private, but also public, post-secondary schools. The Georgetown Center has calculated that kids from families earning $106,000 or more comprise 37 percent of all students in “very selective” public colleges while those from families earning less than $30,000 represent only 18 percent. In open-admission public colleges (the least selective schools) the proportions are almost exactly reversed. That matters because on average, the elite schools spend at least twice as much per student as less selective campuses.

Eliminating public university tuition for most families could encourage more low-income young people to pursue higher education by signaling them from a young age that they can afford it, notes Richard Kahlenberg, an expert on college access at the Century Foundation. But tuition-free college, he says, wouldn’t address the principal reasons the top public schools don’t admit more applicants from disadvantaged backgrounds: their fear of falling in national college rankings if they accept students with lower test scores, and their reluctance to invest in the extra support often required to help such students succeed. That’s money, Kahlenberg says, “not spent on reducing class sizes or other things that would increase your national rankings.”

If anything, ending public tuition could compound those problems. Today about two-fifths of kids from families earning at least $106,000 annually attend four-year public schools, and just over one-fifth attend private schools, the Georgetown center calculates. If tuition is eliminated for families earning up to $125,000, many experts consider it inevitable that more upper middle-class families now choosing private schools would shift to public options. And since test scores track closely with parental education, those affluent kids would often arrive with the credentials all colleges covet—leaving less room for students from lower-income districts that don’t produce as many gaudy scores. That’s despite the fact, Carnevale says, that research shows most students with somewhat lower test scores and grades can succeed even at elite institutions with the right support.

This squeezing-out is already evident in the University of California system, where low-income students are much less likely to attend the most elite campuses (such as Berkeley, San Diego, and Los Angeles) than less competitive options (including Riverside and Merced). “The average GPA of six of our nine campuses is 4.0 and a near perfect SAT,” says Audrey Dow, senior vice-president of the Campaign for College Opportunity, which advocates for low-income California students. “That’s definitely not what we thought our public university would be—this hyper-elite institution that demands perfection.”

Aides say Clinton is aware of these risks, and if elected, would design the tuition-free proposal to require states to enhance racial and class diversity at their public universities. But devising policies to achieve that goal, especially with the Supreme Court’s limits on affirmative action, isn’t easy.

One key may be building more capacity at public universities: that way if a tuition-free policy draws more young people to choose that option, they aren’t just jostling for the existing spaces. Dow is right when she says, “we just have too few spots at our public universities.” That’s especially true compared to the public college building boom that states including New York and California undertook for the baby boomers.

Eliminating tuition at public colleges could be a powerful means of expanding opportunity—and equipping an increasingly diverse America to compete in the global economy. But only if Washington and the states ensure that these new investments don’t merely reinforce old privileges.