Washington (CNN) A group of Republicans are making good on their threat to attempt to force an immigration floor vote in the House -- potentially paving the way for a showdown among proposals to save the Deferred Action for Childhood Arrivals program .

The lawmakers on Tuesday signed what's known as a discharge petition -- a procedural maneuver that can bring legislation to the House floor if it is signed by a majority of House members regardless of whether it has moved through committee, as is traditionally the case for most legislation. If the petition were to pick up enough supporters, it would set up a floor debate on four different immigration measures as early as June.

The move is unusual for members of the majority party, who are effectively going around House Speaker Paul Ryan to set up a vote on legislation that GOP leadership has refused to call to the floor for a vote. Still, the members insist they are making an effort to be deferential to leadership, by leaving one bill open to the speaker's choosing.

The effort is being spearheaded by three moderate Republicans who have long been vocal about trying to save DACA, a program that protected young undocumented immigrants who came to the US as children -- Reps. Will Hurd of Texas, Jeff Denham of California, and Carlos Curbelo of Florida. Curbelo officially introduced the petition Wednesday morning.

In an exclusive interview with CNN, the three moderates said the goal was to have a long overdue immigration debate without a predetermined outcome.

"This institution should be driven by courage, not by cowardice, and the goal should not be to suppress members from pursuing their legislative goals, it should be to empower each member, and that's what we're trying to do," Curbelo said. "The goal is to empower each member of the House, including the speaker, to advance the solution that each member believes is the best one for this challenge and to try to gain supporters for that solution. So this is not in defiance of anyone."

A spokeswoman for Ryan didn't comment specifically on the petition, saying efforts to pass immigration legislation continue in general.

"We continue to work with our members to find a solution that can both pass the House and get the president's signature," AshLee Strong said.

The three members who pitched the proposal held a news conference Wednesday afternoon announcing their move, and were joined by fellow Republican Reps. Mario Diaz-Balart, of Florida, John Faso, of New York; Mia Love, of Utah; and Ileana Ros-Lehtinen, of Florida.

The petition has more than a dozen Republicans on it already, Denham said, and while the members would not reveal their list of supporters in advance, they did note the signatories come from across the ideological spectrum.

Some more conservative members of the House could back the effort because it would allow a vote on a hardline immigration measure they've supported, and circumvent what conservatives often complain is overly restrictive procedural controls by leadership.

"I think you'll see many different caucuses throughout the House that are engaged in this debate that are focused on different solutions," Denham said. "I think it's our job. Congress needs to do its job and be held accountable."

President Donald Trump sought to end the executive program put in place under the Obama administration last September, but a collection of court rulings have found Trump's action likely does not pass legal muster, and kept the program largely in place. A group of r ed states recently sued in a different court to try to have the program itself declared unconstitutional.

Lawmakers have sought to pass legislation that would enshrine the program in law, which would address critics who say it goes beyond the authority of the executive branch. But efforts to protect it in Congress have been unsuccessful.

It remains unclear if the new effort will pick up enough support to force a House showdown. While it starts with a number of Republicans in support, it would still need to roughly double the number of GOP members signing on and pick up all Democrats in the House. Effective discharge petitions have been rare in House history, though not unheard of.

Democrats have insisted on a path to citizenship for DACA recipients and have opposed measures that they say are too aggressive or punitive to immigrants in return, though they have agreed to $25 billion in border security funding. Most Republicans have been split about a path to citizenship, and have insisted any such deal must include cuts to legal immigration and hardline measures to target illegal immigration, as well.

The discharge petition would support what's known as a "queen-of-the-hill" rule, which would bring four competing immigration-related bills to the floor for debate and a vote. Denham and Hurd had previously announced the rule had the support of 50 Republicans and 190 Democrats, more than 20 members over the threshold for a majority of the House, but it's unclear if all of them will back the petition.

Denham, Hurd and Curbelo's move Tuesday would pave the way for a floor vote on a hardline bill from Republican Reps. Bob Goodlatte, Mike McCaul and others that does not include a path to citizenship; a creation of a program like DACA without any border security measures from Democrats; a bill Ryan would offer; and a bipartisan bill from Hurd and California Rep. Pete Aguilar that would pair a path to citizenship with a direction to the administration to gain "operational control" of the border by the best means available. The rule also allows for the authors of the bills to change them, and the members expected all of the proposals would evolve before a floor vote -- especially to include language that would appropriate billions for border security.

GOP leadership has agreed to whip the Goodlatte bill, but it has failed to gain the support of enough Republicans to make it viable to pass the House. The President has backed Goodlatte's legislation and rejected all the other proposals put forth besides his own hardline plan. The reserved spot for the speaker could be any bill of his choosing.

The "queen-of-the-hill" procedure would mimic an exercise in the Senate earlier this year, when votes on four competing immigration proposals ended with none reaching the number of votes necessary to move forward, including the President's plan.

"This debate is too important not to have," said Hurd, who has roughly one-third of the entire southern border as part of his district, more than any other single member. "Let's have this debate on the House floor and let everybody bring their ideas to the forefront."