There is a minimum speed required to roll a vehicle over, without this minimum speed the vehicle lacks the kinetic energy required to go over. For a lightly loaded Ford Explorer this minimum speed is probably about 35 mph on level ground. Heavily loaded vehicles will go over at lower speeds, maybe as low as 30 mph. Lower pressure lowers the vehicle slightly. Lower pressure decreases the steering response and decreases cornering capability.
Lower pressure produces a softer, more car-like ride.
What is clear is that lower pressures increase tire flexing and heat build up in the tires. . So you'll have to be more careful in normal driving. But this seems to be a good trade off in warm climates at least until you get the Firestone tires off the vehicle. This is a problem they inherited from the Bronco II and earlier Ford pickups. But any vehicle can rollover if it leaves the paved surface and the wheels dig in to trip it.