An anti-gun violence group backed by billionaire former New York Mayor Michael Bloomberg has taken its fight against open carry to one of America’s largest supermarket chains following successful campaigns aimed at other retailers.

Moms Demand Action for Gun Sense in America, part of Bloomberg’s Everytown for Gun Safety coalition, launched a petition on Monday calling on Kroger to ban shoppers bringing in firearms to any one of its 2,642 stores.

Similar pushes in recent months have forced companies like Chipotle and Starbucks into making statements against carrying guns into outlets across the U.S. The campaign comes amid a series of open-carry events across America, in which gun rights advocates toting weapons have taken pictures at stores in states where it is legal to brandish a gun in public.

The call to make Kroger guns-free follows a series of shootings at its stores across the country, said Shannon Watts, founder of Moms Demand Action for Gun Sense in America. In one case in June, two people reportedly died in a “double fatal shooting” that turned out to be a murder-suicide at a Kroger in Marietta, Georgia.

“The history of shootings at Kroger stores underscores the risk of allowing open-carry demonstrations and rallies to continue,” Watts said.

Kroger, in an emailed response to Al Jazeera, said that it did not want to task employees with policing customers.

“Millions of customers are present in our busy grocery stores every day, and we don’t want to put our associates in a position of having to confront a customer who is legally carrying a gun,” said Kroger spokesman Keith Dailey.

But even if companies like Kroger respond to campaigns by the gun control advocates, that doesn’t mean it will amount to corporate policy.

In May, Chipotle appeared to bow to pressure from Moms Demand Action for Gun Sense following a #BurritosNotBullets online campaign.

“Historically, we felt it enough to simply comply with local laws regarding the open or concealed carrying of firearms, because we believe that it is not fair to put our team members in the uncomfortable position of asking that customers refrain from bringing guns into our restaurants,” Chipotle spokesman Chris Arnold then told Al Jazeera in an email.

“However,” said Arnold, “because the display of firearms in our restaurants has now created an environment that is potentially intimidating or uncomfortable for many of our customers, we think it is time to make this request.”

Erich Pratt, spokesman of gun rights advocacy group Gun Owners of America, observed that Chipotle’s statement in May as well as Starbucks’ statement following another campaign by gun control advocates in August 2013 was not tantamount to corporate policy.

“Most of [the companies] do seem to be doing the old proverbial splitting the baby in half, or trying to appease both sides,” Pratt said, explaining that companies responding to the campaigns are doing what they can not to anger patrons on either side of the debate.

Amid a campaign to ban guns at Jack in the Box, the hamburger giant released a statement saying it “preferred” patrons leave their guns at home. Target last month also “asked” its customers not to bring guns to stores.

The statements were hailed as successes in national media and by gun control advocates. But gun rights advocates say that where there’s a will to tote guns, there’s always a way.

“We’ll see what Kroger is doing,” Pratt said. “Let’s assume they really do come down with this open-carry ban – they haven’t said, ‘Don’t bring your concealed firearms.’ Even if they do come up with a policy against open carry, they still haven’t gone the whole way with banning them.”