I haven’t heard from Publix on their numbers, but they do have a program for getting you to try their store brand. They select a couple items (national brands), and offer you the Publix store brand for free when you purchase the national brand.
This week you can buy Sunmaid Raisins, McCormick Sloppy Joe Mix and Tyson Chunk White Chicken (canned) and get the Publix brand of these items for free to try.
The deal is only good for this week (July 29 through August 4). Look for the displays at the store.