Senator John Cornyn has signed on to the effort. Photo: Bill Clark/CQ-Roll Call,Inc.

Twelve days ago, there was a horrific shooting at a church in Sutherland Springs, Texas, that left 26 people dead. In a somewhat unusual move, a handful of Republican lawmakers joined Democrats in pledging to fix the error that allowed killer Devin Kelley — who was court-martialed and kicked out of the Air Force for assaulting his wife and infant stepson in 2014 — to obtain a gun.

On Wednesday Senators Chris Murphy, a Democrat, and John Cornyn, the chamber’s second-highest-ranking Republican, announced they’ve teamed up to craft legislation that aims to improve the national background-check system. Their bill, which Democratic senator Richard Blumenthal is also working on, would create financial incentives for states to report offenses to the National Instant Criminal Background Check System (NICS), and also ensure that federal agencies are following through on the existing requirement to send information to the NICS (which the Air Force failed to do Kelley’s case).

“What our bill does is it attempts to fix that both at the federal level and provide additional incentives to the local states,” Cornyn said. “It may be as simple as just getting them to do what they’re already required to do.”

The senators are hoping to attract more Republican support before they formally introduce the bill, and there’s a chance that this could be the rare gun-control measure that the right can get behind. Cornyn has an A+ rating from the National Rifle Association, and the organization is generally in favor of the enforcement of existing background-check requirements over new laws.

Still, there are a mountain of reasons for gun-control advocates to be pessimistic. After 58 people were killed by a shooter in Las Vegas on October 1, there was talk of banning bump stocks, which allow semiautomatic weapons to fire like illegal automatic weapons. But according to the Washington Post, proposals for a bill banning the attachments appear to be going nowhere.

Senators Jeff Flake, Martin Heinrich, and Blumenthal are still working on changing the way the military reports domestic-violence misdemeanors. But there hasn’t been a major push to close all the other loopholes that allow domestic abusers to obtain guns, though another mass shooting was committed in Northern California this week by a man with a history of domestic violence who wasn’t supposed to have guns.