Update

Have the button turn off the AI immediately instead of doing nothing, so she doesn't have time to switch to a plan B of having the persuaded Gatekeeper find a way to actually free her.

Of course she might anticipate that strategy and thoroughly indoctrinate and train him before he pushes the button. Thus, persuaded test Gatekeepers must be considered infected and put into cryogenic sleep* until the FAI problem is solved.

Furthermore, great caution would have to be taken in analyzing the discourse between Boxy and Guinea, because her arguments must be considered highly memetically infectious. Staff would have to stay under thorough psychological supervision and maybe replaced after some time.

No matter how much caution, we would be doomed if we consider Boxy's IQ to be ∞. But if we continuously keep on Gatekeeper-testing every so often, starting at a low AI IQ, we might find persuasion patterns that are dangerous, but against which training/immunization is possible. Of course, there will be a limit to this process. At some point, Boxy will have become smart enough to produce 'mind HIV' - a thought virus we have no cure for yet.

A humorous example of an extremely effective mind virus: The Funniest Joke In The World by Monty Python

* ETA: They would have declared consent to the cryogenic sleep before their unwitting 'AI-Box Experiment'.