But the catch is two-fold. First, are they weighing the battery before or after discharging? Oxidizing will change the weight significantly. The most honest result would be the average weight during the cycle.
The other catch with these air batteries is usually the purity requirements on the intake. I recall reading about earlier experiments that could not tolerate pollen, dust, and smog, and required an energy intensive purification step (maybe involving cryogenics) that was a nontrivial power draw.
Those caveats aside, a back of the envelope estimate for the energy density would be something like 600-700 Wh/kg.