One thing that we have had for this sort of thing is that cards cannot cascade-negate themselves, otherwise you can indeed have loops. So, the BTN part of the character stays active.
In this case, I can give you a better reason. Since they say "while remains," they turn off when they leave play. As an ongoing ability, the BTN triggers once it is no longer negated. It negates the enhancement that removed the placed card, which now returns. Since it was not active, the placed card now tries to activate...but it cannot, because it is prevented. It is not placing itself again, but it is trying to negate, and that has been prevented by the already-active BTN ability.
Note that this works the same even if you only negated the enhancement rather than discarding it.
