Part of What We Don't Know About Sprawl and Obesity

Numerous studies (such as the one referenced here) have suggested that there is some connection between sprawl and obesity, because residents of sprawl walk less and are thus more likely to weigh more.  

However, people in low-income urban areas are more likely to be overweight- even in relatively walkable places.  One hypothesis is that poorer people tend to have poorer diets, and that this effect simply overrides the positive effects of walkability.  An alternative theory is that poorer people actually do walk less because of fear of crime and traffic.  Which hypothesis better fits reality?

It seems to me that to answer the question we would have to find out how much low-income urbanites walk.  Do they walk more than anyone else?  Do they walk more than suburbanites but less than high-income city residents?  Or do they walk as little as suburbanites?   

Similarly, high-income suburban communities tend not to be as obese as the rest of America.  Is this because they get more exercise?  Or is it purely because of diet?  To find out, we'd have to get a better sense of who gets how much exercise, either through walking or by other means.