When Beck is running up the stairs for the second time, after grabbing Joe's keys, the gate is wide open; it's then closed and locked again when the scene changes as she gets to the top of the stairs.
In the final part, when Beck finally leaves the cage and manages to lock down Joe, they have a big fight and you can see multiple times the keys are still on the doors lock. She runs for the stairs to the upper floor and when she returns Joe's gone, saying he used another key he had hidden which wouldn't be possible since the other keys were still there on the lock, as can be seen multiple times before.