Supporters of House Democrats taking part in a sit-in on the House floor shout encouragement from outside the U.S. Capitol on June 22. (Photo by Pete Marovich/Getty Images)

A bipartisan group of House lawmakers is getting behind a compromise gun control proposal that has the support of the majority of the Senate — another sign that moderate Republicans and Democrats are trying to find common ground on the contentious issue.

Reps. Carlos Curbelo (R-Fla.) and Seth Moulton (D-Mass.) on Friday introduced House bill that is identical to legislation crafted by a bipartisan group of senators led by Susan Collins (R-Maine), according to a spokeswoman for Moulton. In a procedural vote Thursday, 52 senators supported the Collins proposal although it would need 60 votes to pass and its unclear when it may be brought up for consideration again.

[Bipartisan gun control compromise fails to nab 60 votes, but survives a first hurdle]

Curbelo and Moulton have the support of some Republicans in close reelection battles, such as like Rep. Bob Dold (R-Ill.), but they face stiff odds of getting a vote for their proposal in the House.

Speaker Paul D. Ryan (R-Wis.) has refused to cede to Democrats’ demands to hold votes on two gun measures following the mass shooting in Orlando earlier this month: One to deny firearms to terrorists and another to expand background checks.

Democrats staged a dramatic sit-in on the House floor this week to push for the votes, but Ryan called the protest a “publicity stunt” and said he would not be pressured into holding votes on measures Republicans believe run counter to the Second Amendment.

He also noted that the bills being pushed by House Democrats were defeated in the Senate earlier this week.

Republicans in the bipartisan House group urged constituents on Friday to contact their representatives and press for action, saying they hope public pressure will compel their colleagues to sign on to the proposal and lead GOP leaders to hold a vote.

Democrats in the group said they expect their members to back the compromise proposal too, even though it is far less expansive than the measures they were demanding votes on during their floor protest.

“We’d so much rather see some progress than no progress at all,” Moulton said. “That was the point of the sit-in.”

The Collins and Curbelo-Moulton proposals would make it possible for the government to deny firearms to anyone whose name appears on the No Fly List or the Selectee List – two subsets of the FBI’s Terrorist Screening Database comprised of about 109,000 individuals, only 2,700 of them United States citizens.

It also contains a “look-back” provision that would alert the FBI whenever someone who had been on one of those lists during the past five years bought a gun. It also gives anyone who thinks they were mistakenly denied a firearm the right to appeal the decision quickly, and have their legal fees covered by the government if they are proven right.

Collins’ proposal survived a test vote in the Senate Thursday, but it failed to garner the 60 votes that will be needed to get over future procedural hurdles. Before Thursday’s vote Collins said she believed there were enough Republicans “on the cusp” of backing her proposal to clear the 60-vote threshold had GOP leaders not scheduled a vote at the same time on another proposal that allowed wavering lawmakers another option to show they supported some restrictions on purchasing a gun.

Democrats declared a minor victory anyway, calling the test vote the first time the Senate had voted against the wishes of the National Rifle Association since 1994, when Congress passed an assault weapons ban, which expired in 2004.

“Even though it wasn’t a big victory, it was a victory,” said Senate Minority Leader Harry Reid (D-Nev.). “Now the Republican leadership has the responsibility to bring this bill to the floor for a real vote.”

It is not clear that GOP leaders in the Senate are interested in giving Collins’ proposal an up-or-down vote.

Democrats in both chambers said they will continue to push for votes on gun control measures.

Though House Democratic leaders disbanded their sit-in after about 26 hours Thursday, they insisted they “will be back,” as Minority Whip Steny Hoyer (D-Md.) said. They would not specify whether they planned to resume a sit-in or try other tactics when the House returns from its Fourth of July recess.