I do not believe that is true. In my hockey travels I was always surprised to see that Chicagoland had lower prices on many staples when I would go out shopping to get milk and cereal for the kid to eat as a snack before early games.
Beer and booze we definitely have had lower prices than neighboring states.
This was typically true because denser areas have economies of scale, but the problem of late is wages. A lot of that goes back to NIMBYism where housing is blocked in wealthier areas that still want to have stores, so retail labor either has had a disproportionate increase in housing costs, or a similar increase in transportation costs because they have to commute in.
With booze, it's because even in states like Tennessee where bourbon is an iconic state product, the government is still run by puritanical evangelicals who tax the living shit out of booze. Illinois has a super relaxed attitude to liquor compared to most of the US.
When I got to UIUC I had to take a double take to see liquor at Walgreens. And I came from Colorado! To this day, Colorado still has some funky laws. Grocery stores can sell beer - that's it. Not even wine. You have to go to a liquor store. For a long time, Colorado had special "3.2% alcohol" beer that grocery stores sold, higher than that you had to go to a liquor store. The craft beer revolution ended that because the grocery stores wanted to carry Fat Tire/etc... not just 3.2% coors. But still no wine. I was making Chicken Marsala, but grocery store can't sell Marsala wine. I had to go to 4 liquor stores to find Marsala, because it's really not something you drink, you just cook with it, so most stores don't carry it.
My first job was in Boston. I lived in Arlington Mass, which was a DRY TOWN. They had few restaurants because they couldn't sell wine in the restaurants, but of course a couple of Italian places had gotten some exemption to allow BYOW. New Hampshire you could buy beer in stores but liquor you had to go to a State run liquor store. Don't even get me started on Utah.