> Using a structural demand model, we find that exposing low-income households to the same food availability and prices experienced by high-income households would reduce nutritional inequality by only 9%, while the remaining 91% is driven by differences in demand.
On the topic more generally
> Dr. Sturm found no relationship between what type of food students said they ate, what they weighed, and the type of food within a mile and a half of their homes.
https://www.nytimes.com/2012/04/18/health/research/pairing-o...
Original study
> School and Residential Neighborhood Food Environment and Diet Among California Youth
> No robust relationship between food environment and consumption is found. A few significant results are sensitive to small modeling changes and more likely to reflect chance than true relationships.
https://www.ajpmonline.org/article/S0749-3797%2811%2900849-X...
In the case of a food desert within a residential area, it's probably a lot easier to boost demand by artificially increasing supply, rather than the other way around.