Seems like a fairly weak correlation. Unfortunately, they don't have any meaningful descriptive statistics, even in the full Pew report that is linked in the article. What's different about DC and Atlanta, for example? Why are they so immobile? I would guess public transport plays into it, but both of those places have way better transit systems than Detroit or LA.
I'm actually surprised my fair city of Detroit isn't at the top (bottom?) of the heap of immobility. I'm not sure who the people who are pulling themselves out of poverty around here are. It all seems fairly static in the city and the 'burbs. Entrepreneurship is almost impossible in America's ghettos, because even if you have an idea and the drive to pull it off, there's nobody who can afford your product or service. Also, mobility used to be primarily driven by manufacturing jobs. High pay, low skill jobs have gone the way of the buffalo, so what's a poor, undereducated person to do? The whole problem of the ghetto has become a self-sustaining, self-defeating death spiral. I don't know if it's possible in principle to stop it.