I'd be curious to see the logistics of that theory (not challenging you, just genuinely curious). If the grocery stores are all chains, I don't see why
they would make the distinction of which location gets the fresh produce. And, though I don't like to admit it, you'd think the opposite would be true - that chains would invest their highest-quality produce in the place with the customers who have more buying power.
If it was instead a regulations thing at the city or state level, that seems fiddly and hard to legislate/enforce. How would a government verify that stores in low-income neighborhoods are getting the best produce? And, more tongue-in-cheek, why have I never heard rich people complaining about it?
My first guess would be that low-income families often have less free time, which means less cooking time, which means more prepackaged or fast food. That's a long-held belief that I've heard regurgitated pretty frequently, though a (quick, shallow) Google search tells me that equation isn't as clear-cut as I thought and home-cooked meals don't cleanly break along income lines.