It would only be unfalsifiable if the connection was one-way, i.e. our souls receive input from the world but have no effect on the world. We would effectively be passengers in a meat puppet, and any influence we have on that puppet is purely imaginary. Certainly possible, but not usually how it's conceived, and then you have the question of how the meat puppets got the idea of consciousness and souls in the first place.

If souls can influence the world, as is the common belief for souls, then in principle that influence or its absence can be detected.