(CNN) On Wednesday, two more Republicans signed on to a petition that would force an open-ended and unpredictable immigration debate onto the House floor -- a nightmare scenario that the White House and GOP leaders seem, at least at the moment, unable to stop.

The decisions by Republican Reps. John Katko of New York and Dave Trott of Michigan to sign on to the petition came in the wake of private urgings from House Speaker Paul Ryan to his GOP colleagues not to add their names to the list. The duo's act of defiance brought the number of Republicans signed onto the effort to 20, just five short of the number they need to force votes on four competing measures aimed at preserving the Obama-era Deferred Action for Childhood Arrivals (DACA) program. (That math -- 25 Republicans -- is dependent on all 193 Democrats supporting the effort.)

At the heart of the debate is the long-standing division within the GOP conference over what to do on immigration reform -- and the popular DACA program in particular -- amid swirling political winds.

On one end of the debate stand Trump, Ryan and the rest of the House Republican leaders who view a wide-open floor debate -- and a series of roll-call votes -- as a potentially disastrous visual for the party and one without any likely positive result. Ryan called such efforts "futile" on Thursday.

Trump, who ran hard on the need to toughen US immigration policy by, among other things, building a wall on the southern border, has little interest in some sort of compromise immigration legislation crafted by two dozen Republicans and congressional Democrats. He has said in recent days that he will push for full funding of his border wall in the next budget negotiation and has even floated the possibility he would be willing to shut down the federal government if his demands are not met.

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