With the White House withdrawing from a demand for funding for President Donald Trump’s border wall, some of the nation’s fiercest immigration hawks are warning the president not to back away from his most recognizable campaign promise.

Trump had requested a $1.4 billion down payment on the wall in a spending bill that must pass by the end of the week to keep the government open. But White House counselor Kellyanne Conway told Fox News on Tuesday that Trump would not demand it.

“The wall is a very powerful symbol of whether Trump can get anything done in this town.”

“Building that wall and having it funded remains an important priority to him,” she said. “We also know that can happen later this year and into next year.”

But Trump tweeted that the wall would be built and a White House aide later told CNN that the president expects the spending bill will have money for it.

The waffling has frustrated organizations that want tighter border enforcement.

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“We don’t expect there will ever be a wall,” said William Gheen, founder of the Americans for Legal Immigration political action committee. “We think that he lied to all of us.”

Dan Stein, president of the Federation for American Immigration Reform (FAIR), told LifeZette that starting the wall project has importance beyond the impact it could have on border crossings.

“Trump campaigned on a border wall as a pillar, a marquee, a cornerstone of his campaign,” he said. “There damn well better be a wall … The wall is a very powerful symbol of whether Trump can get anything done in this town.”

FAIR last week paid to promote a #buildthewall hashstag on Twitter. Stein, who remains supportive of the president’s immigration goals, said Republicans should not fear a government shutdown if it is forced by Senate Minority Leader Chuck Schumer (D-N.Y.) and other Democrats.

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“Why is it every time there’s a government shutdown, Republicans assume they’ll take the blame?” he asked. “Chuck Schumer and the Democrats are handing Republicans an amazing political issue.”

Other organizations favoring more aggressive immigration enforcement, however, did not depict the wall as a make-or-break issue.

“It’s not our top priority,” said Chris Chmielenski, director of content and activism at NumbersUSA. “We would like to see completion of the Secure Fence Act.”

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That is a reference to a 2006 law calling for a 700-mile long, double-layered fence. Although Congress passed the bill, it never provided enough money to complete the project.

Beyond that, Chmielenski said, the Trump administration’s focus should be on hiring more border patrol and Immigration and Customs Enforcement officers and improving technology. He said Trump also should push Congress to require employers to use the E-Verify system, which instantly checks the background of all new potential employees to verify their legal status to work.

“But anything that helps secure the border, we’re obviously going to be for,” he said.

Joseph Guzzardi, a spokesman for Californians for Population Stabilization, said his organization opposes a border wall because of environmental concerns. He said the administration should work to speed up a planned crackdown on so-called “sanctuary” jurisdictions and other measures that would deter illegal immigration in the first place.

“President Trump understands he doesn’t want the bad vibe of a government shutdown for which Republicans would get blamed,” he said. “I would urge him to get busy on E-Verify.”