THE LARGEST crowd of demonstrators in years gathered outside the Supreme Court on April 18th over United States v Texas, a case in which 26 states are challenging Barack Obama’s executive orders to relieve 4.3m undocumented immigrants of the immediate threat of deportation. Waving American flags and holding signs reading “Keep Families Together”, “We Belong Together” and “Families Fight Back” the activists’ message was clear: the justices should uphold Mr Obama’s 2014 orders designed to prevent immigrant families from being torn apart. Under the programme known as Deferred Action for Parents of Americans and Lawful Permanent Residents (DAPA), parents whose children have American citizenship or a green card would be eligible to apply for protection from deportation and authorisation to work if they agree to background checks and can prove they’ve been in the country since 2010. An undocumented alien with deferred-action status would be able to work and pay taxes and would not live in fear of being separated from his children if, during a routine traffic stop, his unlawful presence were discovered.

Judging from the conservative justices’ aggressive questioning of Donald Verrilli, the lawyer defending this White House policy, it appears that only the four more liberal members of the Supreme Court are ready to vote to uphold Mr Obama’s moves against claims of executive overreach. Four is one vote short in this case, since a tie among the eight justices (reduced from nine in February, when Antonin Scalia died) would automatically affirm the lower court ruling that short-circuited DAPA’s implementation. But the best-case scenario for DAPA advocates—in which Chief Justice John Roberts or Justice Anthony Kennedy, or both, vote with the liberals to turn back the 26 states’ challenge—leaves question marks for immigrant families.

Revealing themselves to federal authorities could bring immigrants opportunities, but the decision to apply for DAPA entails risks as well. Before the injunction against DAPA was put in place, the White House sought to reassure potential applicants by noting that presidents typically honour the executive orders of their predecessors. But both Donald Trump and Ted Cruz, the most likely Republican nominees for president, pledge to cancel Mr Obama’s orders if they win the White House. Both candidates support policies that involve detaining or deporting all of America’s 11.3m undocumented immigrants. This means that provisional protection under DAPA would probably evaporate if a GOP candidate won in November, and immigrants who have made themselves known to immigration officials may rue the day they came out from the shadows.

After the ruling is handed down in June, according to Beth Werlin of the American Immigration Council, DAPA (if upheld) may take several months to launch fully. It is unlikely, she says, that the programme will be properly underway by November, giving immigrants a chance to weigh their chances under the next administration. Marielena Hincapié, executive director of the National Immigration Law Centre, agrees that a Democratic win in the presidential election would encourage a surge of applicants in the autumn and in early 2017. But she doubts everyone will wait. In 2012, Ms Hincapié notes, thousands of applications for a similar programme protecting immigrant children poured in during the late summer and early autumn—before anyone knew whether Mitt Romney and his policy of “self deportation” would find a home in the White House. Activists like the 5,000 people who demonstrated in front of the Supreme Court this week, Ms Hincapié predicted, may jump to apply as soon as the registration process is open. She says that “advocates across country are already starting to prepare” for the day DAPA gets the green light.