(CBS).– Publix is adding its name to the growing list of retailers asking patrons to leave their firearms behind when shopping. The supermarket chain is following CVS, Kroger, Walgreens, Walmart and other large chains in requesting that customers refrain from openly toting guns in their stores, even if they live in states that allow the practice.

Lakeland, Florida-based Publix “respectfully requests that only law enforcement officials openly carry firearms in our stores,” a spokesperson told the South Florida Sun Sentinel. The grocer operates in seven states, including Alabama and North Carolina, where openly displaying weapons is legal. The company did not immediately return a request for comment.

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After the mass shooting at a Florida high school in 2018, some surviving students called for a “boycott of Publix after the disclosure the regional chain made political donations to Adam Putnam, at the time a candidate for governor and staunch supporter of the National Rifle Association.

The most recent burst of changes in “open carry” policies started with Walmart and Kroger. Walmart, the world’s biggest retailer by revenue, had been under intense pressure after separate shootings that recently killed 24 people in two of its stores within a week.

In announcing the new open-carry policy, Walmart CEO Doug McMillon described multiple incidents of people “attempting to make a statement” by entering a Walmart wielding firearms, scaring workers and customers. Walmart has also stopped carrying some types of ammunition that have been used in mass shooting.

Despite such shifts, retailers seem to be walking a fine line between trying to respond to the rash of mass shootings of late while not angering customers who support gun rights, experts said, noting that the new policies stop short of outright bans on firearms in stores.

“Most stores are nervous about this – gun carriers don’t take kindly to being told” not to bring their guns along, said Timothy Lytton, a law professor at Georgia State University and firearm industry expert. Given Walmart’s nationwide customer base, the retailer is “worried about offending pretty much anybody,” he added.

Most U.S. states allow residents to openly carry firearms in public spaces, but the rules vary. By contrast, guns can be prohibited on private property.

Texas, like many open-carry states, stipulates that retailers and other private-property owners that want to ban firearms may do so. The catch: They “have to post this big ugly sign saying so,” said Houston attorney Patrick Luff, whose firm is representing a family in which both parents were shot and injured in last month’s mass shooting in a Walmart store in El Paso, Texas. That signage requirement typically deters businesses from enacting outright bans, because “then you have to openly announce to your Texas customers that you do not support open carry,” he said.

Meanwhile, despite the recent move by retailers to change their gun policies, they are not paving new ground. Back in 2013, then-Starbucks CEO Howard Schultz made his “respectful request” that customers not carry firearms in the chain’s restaurants. Target followed a year later, with the company’s then-interim CEO stating that “bringing firearms to Target creates an environment that is at odds with the family-friendly shopping and work experience we strive to create.”

While none of the major retail, supermarket and drugstore chains that released open-carry announcements this week made clear exactly how they would deal with gun owners who ignored their requests, Walmart said it would take a “very non-confrontational approach.”

Luff expressed doubt that the new policy would have much impact at Walmart. “The simplest question is, ‘What’s different yesterday that wasn’t true on August 2, the day before this happened?’ ” the lawyer asked, referring to the August shooting in El Paso.

Beyond facing conflicting desires of their customers, retailers also risk mounting legal exposure due to mass shootings. Walmart reached at least two legal settlements with families who lost loved ones in a 2014 mass shooting in Overland Park, Kansas, outside a Jewish community center. And Kroger was sued last month by the daughter of a man shot to death at one of its grocery stores, with the suit calling on the chain to ban patrons from carrying firearms in its stores.