Will the legislative majorities at the Statehouse join fellow Republican Mike DeWine in advancing public safety through sensible gun regulation? The case has gained strength this week. According to news accounts, the gunman who killed seven people in a mass shooting last weekend in Odessa, Texas, purchased his weapon via a private sale. Thus, he took advantage of a glaring loophole in the background check system.

The news reports add the gunman took this step after failing a background check.

This is the weakness the governor seeks to address. He has proposed requiring background checks for all firearm sales in Ohio, with limited exceptions. The provision would look to cover the roughly one-fifth of gun sales that go forward without a background check.

Would such an expansion reduce substantially gun violence? Probably not. Yet, as the governor explained to reporters last week, the purpose isn’t to find a cure-all. The objective is to adopt a range of measures that as a whole provide better protection for the public without jeopardizing individual gun rights. At one point during the press conference, Jon Husted, the lieutenant governor, reminded that the gunman who killed 26 people in Sutherland Springs, Texas, two years ago exploited a similar gap in the law.

Count 33 lives that might have been saved with an improved background check system.

Ohioans for Gun Safety made the connection in a press release highlighting this news about the Odessa shooting. The advocacy organization is seeking to place before Ohio voters a ballot issue that would close the private sale loophole. Polls show 90 percent of Ohioans in support, including 87 percent of gun owners. The politics are obvious: Lawmakers can seize the initiative, or they can let voters take charge, the ballot issue virtually certain to pass.

If there is doubt about such an outcome, it narrowed most likely with word from Walmart. The country’s largest retailer took time to think about how to respond to the mass shooting at its store in El Paso, Texas, last month, in which a gunman killed 22 people. On Tuesday, it announced it will stop selling ammunition that can be used in military-style assault weapons. It will halt sales of any ammunition for handguns.

Walmart also called for expanded background checks and urged consideration of a revived assault-weapons ban. The company has taken similar stands in the past, ending the sale of handguns in the 1990s and raising to 21 the minimum age for purchasing a gun. Yet this time must be different, as its chief executive urged in a statement: “As we’ve seen before, these horrific events occur and then the spotlight fades. We should not allow that to happen. Congress and the administration should act.”

On Capitol Hill, that translates to the dim prospect of Mitch McConnell, the Senate majority leader, permitting his chamber to take up gun-safety legislation approved in the House, including appropriately expanded background checks. At the Statehouse, the key players are Larry Obhof, the Senate president, and Larry Householder, the House speaker, each leading a caucus long friendly to the wishes of the National Rifle Association.

Then, Dayton happened, a gunman killing nine people in less than a minute, just hours after the El Paso massacre. Now, the Walmart chief executive sounds like Nan Whaley, the Dayton mayor, both saying: Do something. They are joined by Mike DeWine and Jon Husted, who have taken the lead in pushing for better background checks and other steps, practical, balanced and effective, or what should win the support of Republican legislative leaders.