Any Republican who decides to back the legislation can argue that they just want existing laws to be enforced. And it looks like the GOP won’t have to fear backlash from the gun lobby. “We applaud Senator John Cornyn’s efforts to ensure that the records of prohibited individuals are entered into NICS,” Chris Cox of the NRA said in a statement. “The National Rifle Association has long supported the inclusion of all legitimate records in the National Instant Criminal Background Check System.” The National Shooting Sports Foundation, a trade association for the firearms industry, put out a statement on Thursday in which it “praised U.S. Sen. John Cornyn (R-Texas) for his leadership” on the bill.

Gun-control advocates support the bill too, and say it’s evidence that common ground between Republicans and Democrats in the gun debate is possible. “This is both parties affirming that there are people that we believe should not have access to guns, and we want to make sure that the system is set up in such a way that we prevent access to guns for those people,” Christian Heyne, the legislative director of the Coalition to Stop Gun Violence, which supports the legislation, said in an interview. “This is a real, genuine effort from people who couldn’t be further from each other on the other side of the aisle.”

In addition to Senators Murphy, Cornyn, Hatch, Scott, and Heller, Democratic Senators Richard Blumenthal, Dianne Feinstein, and Jeanne Shaheen are also co-sponsors.

But bipartisan support is still no guarantee that the legislation will actually move forward in Congress or ever be enacted. For that, it needs the support of the Republican congressional leadership. In response to a request for comment asking if Senate Majority Leader Mitch McConnell has plans to advance the bill, a spokesman for McConnell said, “We’re reviewing it.”

There are countless examples of a debate over guns flaring up in Congress in response to a mass shooting, only to stall out not long after. Some Republicans in Congress expressed a willingness to consider a ban on “bump stocks,” a device that enables semi-automatic weapons to fire faster, after 58 people were killed in Las Vegas, Nevada, in what has been called the worst mass shooting in modern U.S. history. In mid-October, Republican Representative Carlos Curbelo and Democratic Representative Seth Moulton introduced a bill to ban the use of bump stocks. Weeks later, the legislation had stalled.

Legislation doesn’t always stall out though: In the aftermath of the Virginia Tech shooting in 2007, Republicans and Democrats came together to pass a bill to improve the federal background-check system after it was discovered that the shooter had a history of mental-health problems that should have barred him from buying a gun. The NICS Improvement Amendments Act, a measure similarly intended to strengthen the background-check system, was later signed into law by Republican President George W. Bush.