Asked. Not prohibited?
Very similar to the Walmart policy. No force of law.
You are able to shop at Walmart or Publix at their pleasure. You have no right to be there.
Since it's private property, they can legally ask you to leave. They don't have to give a reason.
If you refuse, they can call the police and have you trespassed.
If you still refuse to leave, or return after having been trespassed, you can be arrested.